Trigonometric Identities: List of Trigonometric Identities
by Menaga K
Updated Feb 03, 2023
What are Trigonometric identities?
Trigonometric identities are mathematical equations that involve trigonometric functions such as sine, cosine, and tangent. They are used to simplify and manipulate expressions involving these functions, and can be derived from the definitions of the functions, the angle sum and difference identities, and the double angle and half angle identities. Some examples of trigonometric identities include the Pythagorean identity, the reciprocal identities, and the quotient identities.
Trigonometric identities are equations that involve trigonometric functions such as sine, cosine, and tangent. These identities hold for all values of the angle and are useful in simplifying and manipulating expressions or equations that involve trigonometric functions. These ratios are defined in terms of the sides of a right triangle, such as the adjacent side, opposite side, and hypotenuse. These identities are derived using the properties of right triangles and are valid for all values of the angle in the domain of the functions.
List of Trigonometric identities
Trigonometry features several identities utilized to address multiple trigonometric problems. These trigonometric identities or formulas can effectively solve complex questions. Let's examine all the basic trigonometric identities.
Pythagorean Trigonometric identities
The Pythagorean identities are a set of trigonometric identities named after the ancient Greek mathematician Pythagoras. They are used to relate the sine and cosine of an angle in a right triangle. The two identities are:
sin²x + cos²x = 1
1 + tan²x = sec²x
The first identity states that the square of the sine of an angle plus the square of the cosine of the angle is equal to 1. This identity can be proven by using the fact that in a right triangle, the square of the length of the hypotenuse (c²) is equal to the sum of the squares of the lengths of the other two sides (a² + b²). If we let x be the angle opposite side a, then the length of side a is a = sin x, the length of side b is b = cos x, and the length of the hypotenuse c is 1. So, a² + b² = sin²x + cos²x = 1.
The second identity states that 1 plus the square of the tangent of an angle is equal to the square of the secant of that angle. This identity can be derived from the first identity by dividing both sides of the equation by cos²x, which leaves us with tan²x = (sin²x/cos²x) and sec²x = (1/cos²x)
These identities are useful in trigonometry and geometry, especially when working with triangles and solving trigonometric equations.
Reciprocal Trigonometric identities
The reciprocal identities are a set of trigonometric identities that express the reciprocal of the trigonometric functions in terms of the other functions. The three reciprocal identities are:
cot x = 1 / tan x
cos x = 1 / sin x
sec x = 1 / cos x
The first identity states that the reciprocal of the tangent function is the cotangent function. This identity can be derived by using the quotient identity, which states that tan x = sin x / cos x. To find the reciprocal of this identity, we divide both sides by cos x, resulting in cot x = 1 / tan x = cos x / sin x.
The second identity states that the reciprocal of the sine function is the cosecant function. This identity can be derived by taking the reciprocal of the Pythagorean identity, sin²x + cos²x = 1. We divide both sides by sin²x, resulting in 1/sin x = csc x = (sin²x + cos²x) / sin x = 1 / sin x + cos x / sin x = cos x + cot x * cot x.
The third identity states that the reciprocal of the cosine function is the secant function. This identity can be derived by taking the reciprocal of the Pythagorean identity, sin²x + cos²x = 1. We divide both sides by cos²x, resulting in 1/cos x = sec x = (sin²x + cos²x) / cos x = tan x * sec x.
These identities are useful in trigonometry and geometry, especially when working with triangles and solving trigonometric equations.
Quotient Trigonometric identities
The quotient identities are a set of trigonometric identities that express the quotient of the trigonometric functions in terms of the other functions. The two identities are:
tan x = sin x / cos x
cot x = cos x / sin x
The first identity states that the tangent of an angle is the ratio of the sine of the angle to the cosine of the angle. This identity can be derived by using the definitions of the sine and cosine functions, which are the ratios of the opposite side and hypotenuse, to the adjacent side of a right triangle.
The second identity states that the cotangent of an angle is the ratio of the cosine of the angle to the sine of the angle. This identity can be derived by using the reciprocal identity of the tangent function, cot x = 1/tan x = cos x / sin x.
These identities are useful in trigonometry and geometry, especially when working with triangles and solving trigonometric equations. They allow us to express the trigonometric functions in terms of the other functions, which can make them easier to work with. Additionally, these identities are also used to find the missing side or angle of a right triangle.
Co-function Trigonometric identities
The co-function identities are a set of trigonometric identities that express the sine and cosine functions in terms of each other. The two identities are:
sin (90° - x) = cos x
cos (90° - x) = sin x
The first identity states that the sine of the complement angle (90° - x) is equal to the cosine of the original angle x. This identity can be derived by using the fact that the sine and cosine functions are periodic with a period of 360°, and that the sine and cosine functions of an angle and its complement angle are complementary.
The second identity states that the cosine of the complement angle (90° - x) is equal to the sine of the original angle x. This identity can be derived by using the same idea as the first identity and the fact that the sine and cosine of any angle sum to 1.
These identities are useful in trigonometry and geometry, especially when working with triangles and solving trigonometric equations. They can be used to find the value of a trigonometric function if the value of its complement function is known. They also allow us to relate the sine and cosine functions to each other and to express one in terms of the other, which can make them easier to work with.
Double angle Trigonometric identities
The double angle identities are a set of trigonometric identities that express the trigonometric functions of twice an angle in terms of the trigonometric functions of the original angle. The two identities are:
sin 2x = 2 sin x cos x
cos 2x = cos²x - sin²x
The first identity states that the sine of twice an angle is equal to twice the product of the sine and cosine of the original angle. This identity can be derived by using the angle addition formula for sine, which states that sin(x + x) = sin x cos x + cos x sin x. By simplifying this equation, we get sin 2x = 2 sin x cos x.
The second identity states that the cosine of twice an angle is equal to the square of the cosine of the original angle minus the square of the sine of the original angle. This identity can be derived by using the angle addition formula for cosine, which states that cos(x + x) = cos²x - sin²x.
These identities are useful in trigonometry and geometry, especially when working with triangles and solving trigonometric equations. They allow us to express the trigonometric functions of twice an angle in terms of the trigonometric functions of the original angle, which can make them easier to work with and simplify the calculations. They are also useful in solving trigonometric equations, especially when the equations include terms involving twice an angle.
Half angle Trigonometric identities
The half-angle identities are a set of trigonometric identities that express the trigonometric functions of half an angle in terms of the trigonometric functions of the original angle. The two identities are:
sin (x/2) = ±√[(1-cos x)/2]
cos (x/2) = ±√[(1+cos x)/2]
The first identity states that the sine of half an angle is equal to the square root of the quantity (1-cos x)/2 with a positive and negative sign. This identity can be derived by using the double angle formula for sine, sin 2x = 2 sin x cos x and then dividing both sides by 2 and taking the square root of both sides.
The second identity states that the cosine of half an angle is equal to the square root of the quantity (1+cos x)/2 with a positive and negative sign. This identity can be derived by using the double angle formula for cosine, cos 2x = cos²x - sin²x and then dividing both sides by 2 and taking the square root of both sides.
These identities are useful in trigonometry and geometry, especially when working with triangles and solving trigonometric equations. They allow us to express the trigonometric functions of half an angle in terms of the trigonometric functions of the original angle, which can make them easier to work with and simplify the calculations. They are also useful in solving trigonometric equations, especially when the equations include terms involving half an angle.
Product-to-sum Trigonometric identities
Product-to-sum identities are mathematical identities that allow the product of two trigonometric functions to be expressed as a sum of trigonometric functions. These identities are particularly useful in calculus and physics when working with trigonometric functions.
Examples of product-to-sum identities include:
sin(x) * cos(y) = (1/2)(sin(x+y) + sin(x-y))
cos(x) * cos(y) = (1/2)(cos(x-y) + cos(x+y))
sin(x) * sin(y) = (1/2)(cos(x-y) - cos(x+y))
There are many different product-to-sum identities, each involving different combinations of trigonometric functions. These identities can be derived using the trigonometric identities and the addition and subtraction formulas for sines and cosines.
Sum-to-product Trigonometric identities
Sum-to-product identities are trigonometric identities that express the sum of trigonometric functions in terms of their product.
For example, the identity for the sum of sines is:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
Similarly, the identity for the sum of cosines is:
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
These identities can be used to simplify trigonometric expressions and to solve trigonometric equations.
This is not an exhaustive list, but rather some of the most commonly used identities.
Supplementary Angles Trigonometric Identities
Trigonometric identities involving supplementary angles are mathematical equations that relate the trigonometric functions of two angles that add up to 180 degrees. Some common examples include:
Each identity can be proven using algebraic manipulation or geometry. For example, the Sine Identity can be proven by drawing a right triangle and labeling the angle opposite the side of length y as x. Using the definition of sine, sin(x) = y/r. Now imagine the triangle is rotated by 180 degrees, the angle opposite the side of length y is now 180 - x and the sine of this angle is also y/r, which is equal to sin(x).
The cosine and tangent identities can be proven in a similar manner by using the definitions of cosine and tangent and observing how they change when the angle is rotated by 180 degrees.
It is important to note that these identities are only true for angles between 0 and 180 degrees.
Trigonometric Identities Proofs
Trigonometric identities are mathematical equations that involve trigonometric functions such as sine, cosine, and tangent. Some common examples include:
Pythagorean Identity: cos^2(x) + sin^2(x) = 1
Quotient Identity: tan(x) = sin(x) / cos(x)
Reciprocal Identity: csc(x) = 1 / sin(x), sec(x) = 1 / cos(x), cot(x) = 1 / tan(x)
Each identity has its own proof, which can involve algebraic manipulation, geometry, or other methods. For example, the Pythagorean Identity can be proven by drawing a right triangle and labeling the lengths of the legs as x and y, and the length of the hypotenuse as r. Using the definitions of sine and cosine, sin(x) = y/r and cos(x) = x/r. Squaring each of these equations and adding them together results in x^2/r^2 + y^2/r^2 = 1, which simplifies to x^2 + y^2 = r^2, the Pythagorean theorem.
It is important to note that these identities are true for all values of x in the domain of the trigonometric functions.
What are the laws of trigonometry?
The laws of trigonometry are mathematical formulas and identities that relate the ratios of the sides of a right triangle to the angles opposite or adjacent to them. The most common trigonometric functions are sine (sin), cosine (cos), and tangent (tan), which are defined in terms of a right triangle as follows:
The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
There are also other trigonometric functions such as cosecant, secant and cotangent which are the reciprocal of sine, cosine and tangent respectively.
Additionally, there are several important trigonometric identities that can be used to simplify equations and solve problems. Some examples include:
The Pythagorean identity: sin²θ + cos²θ = 1
The reciprocal identities: cscθ = 1/sinθ , secθ = 1/cosθ, cotθ = 1/tanθ
The quotient identities: tanθ = sinθ/cosθ , cotθ = cosθ/sinθ
The double angle identities : sin(2θ) = 2sinθcosθ , cos(2θ) = cos²θ - sin²θ, tan(2θ) = (2tanθ)/(1-tan²θ)
These are a few of the many laws of trigonometry that can be used to solve problems.
How to solve trigonometric questions?
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. To solve trigonometric questions, you can use the definitions and identities of trigonometric functions, such as sine, cosine, and tangent, as well as the Pythagorean theorem and the properties of right triangles. You can also use trigonometric ratios to find missing angles and side lengths in a triangle. Additionally, you can use trigonometric equations to solve real-world problems, such as finding the height of a building or the distance between two points.
Solved Questions on Trigonometric Identities
1. (1 - sin A)/(1 + sin A) = (sec A - tan A)2
Solution
L.H.S = (1 - sin A)/(1 + sin A)
= (1 - sin A)2/(1 - sin A) (1 + sin A),[Multiply both numerator and denominator by (1 - sin A)
= (1 - sin A)2/(1 - sin2 A)
= (1 - sin A)2/(cos2 A), [Since sin2 θ + cos2 θ = 1 ⇒ cos2 θ = 1 - sin2 θ]
= {(1 - sin A)/cos A}2
= (1/cos A - sin A/cos A)2
= (sec A – tan A)2 = R.H.S. Proved.
2. cos θ/(1 - tan θ) + sin θ/(1 - cot θ) = sin θ + cos θ
Solution
L.H.S = cos θ/(1 - tan θ) + sin θ/(1 - cot θ)
= cos θ/{1 - (sin θ/cos θ)} + sin θ/{1 - (cos θ/sin θ)}
= cos θ/{(cos θ - sin θ)/cos θ} + sin θ/{(sin θ - cos θ/sin θ)}
= cos2 θ/(cos θ - sin θ) + sin2 θ/(cos θ - sin θ)
= (cos2 θ - sin2 θ)/(cos θ - sin θ)
= [(cos θ + sin θ)(cos θ - sin θ)]/(cos θ - sin θ)
= (cos θ + sin θ) = R.H.S. Proved.
3. Prove that, √{(sec θ – 1)/(sec θ + 1)} = cosec θ - cot θ.
Solution:
L.H.S.= √{(sec θ – 1)/(sec θ + 1)}
= √[{(sec θ - 1) (sec θ - 1)}/{(sec θ + 1) (sec θ - 1)}]; [multiplying numerator and denominator by (sec θ - l) under radical sign]
= √{(sec θ - 1)2/(sec2 θ - 1)}
=√{(sec θ -1)2/tan2 θ}; [since, sec2 θ = 1 + tan2 θ ⇒ sec2 θ - 1 = tan2 θ]
= (sec θ – 1)/tan θ
= (sec θ/tan θ) – (1/tan θ)
= {(1/cos θ)/(sin θ/cos θ)} - cot θ
= {(1/cos θ) × (cos θ/sin θ)} - cot θ
= (1/sin θ) - cot θ
= cosec θ - cot θ = R.H.S. Proved.
4. From the given figure, find tan P – cot R.
Solution:
From the given,
PQ = 12 cm
PR = 13 cm
In the right triangle PQR, Q is right angle.
By Pythagoras theorem,
PR2 = PQ2 + QR2
QR2 = (13)2 – (12)2
= 169 – 144
= 25
QR = 5 cm
tan P = QR/PQ = 5/12
cot R = QR/PQ = 5/12
So, tan P – cot R = (5/12) – (5/12) = 0
5. Find the correct trigonometric identity.
a) tan2θ = sec2θ – 1
b) tan2θ + sec2θ = 1
c) tan2θ – sec2θ = 1
d) tan2θ = sec2θ + 1
Answer: a
Explanation:
The appropriate trigonometric identity used here is sec2θ – tan2θ = 1.
-tan2θ = 1 – sec2θ
tan2θ = sec2θ – 1
6. Evaluate cosec θ sec θ.
a) cos θ + tan θ
b) cos θ – tan θ
c) tan θ – cot θ
d) cot θ + tan θ
Answer: d
Explanation:
cosec θ sec θ = 1sinθ1cosθ
= 1sinθcosθ(cos2θ + sin2θ)
= cosθsinθ+sinθcosθ
= cot θ + tan θ
7. (1 - cos θ)(1 + cos θ)(1 + cot2θ) = 1
Solution :
Let A = (1 - cos θ)(1 + cos θ)(1 + cot2θ) = 1 and B = 1.
A = (1 - cos θ)(1 + cos θ)(1 + cot2θ)
A = (1 - cos2θ)(1 + cot2θ)
Because sin2θ + cos2θ = 1, we have
sin2θ = 1 - cos2θ
Then,
A = sin2θ ⋅ (1 + cot2θ)
A = sin2θ + sin2θ ⋅ cot2θ
A = sin2θ + sin2θ ⋅ (cos2θ/sin2θ)
A = sin2θ + cos2θ
A = 1
A = B (Proved)
8. Verify the following equivalency using the even-odd identities:
(1+sinx)[1+sin(−x)]=cos2x
Solution
Working on the left side of the equation, we have
(1+sinx)[1+sin(−x)]=(1+sinx)(1−sinx)
Since
sin(−x) =−sinx=1
[5pt] = 1−sin2x
[5pt] = cos2x
cos2x = 1−sin2x
9. For a right angle triangle ABC right angle at C for which angle BAC = θ and sinθ = 4/5 Find the value of cosθ.
Solution:
Using the identity, sin2θ + cos2θ = 1
Putting value of sin in the identity,
(4/5)^2 + cos2θ =1
cosθ =√1-(4/5)^2 = 3/5
10. Find the exact value of sin 75° using the trigonometric identities.
Is is known that 75° = 30° + 45°
Apply the sum trigonometric identity of sin to find the value of sin 75°.
sin (A+B) = sin A cos B + cos A sin B
Substitute A = 30° and B = 45° here on both sides:
sin (30° + 45°) = sin 30° cos 45°+ cos 30° + sin 45°
sin 75° = (1/2)⋅(√2/2) + (√3/2)⋅(√2/2)
sin 75° = (√2 + √6)/4 = (√3 + 1)2√2
Here, the values of sin 30°, cos 45°, cos 30°, and sin 45° can be obtained by using the trigonometric table.
Answer: Sin 75°= (√3 + 1)/2√2
Practice Questions:
- Prove the trigonometric identity sin θ cos θ (tan θ + cot θ) = 1
- Prove sec θ √(1 - sin2θ) = 1
- Prove cot θ + tan θ = sec θ csc θ
- What is 34 - 10 + (1 + cos (180))/2?
- Prove tan2 x + 1 = sec2 x
Trigonometric Identities - FAQs
There are several methods for solving trigonometric identities easily, use any of the three trigonometric identities such as reciprocal identities, quotient identities or pythagorean identities. Practice solving trigonometric identities, and memorize common identities and formulas.
There are several types of trigonometric identities, Reciprocal identities, Quotient identities, Pythagorean identities, Co-function identities, Sum-to-Product identities, Double-angle identities, Half-angle identities.
Pythagorean identities are trigonometric identities that are named after the Greek mathematician Pythagoras. They are identities that involve the relationship between the sine, cosine, and tangent functions.
Co-function identities are trigonometric identities that relate the sine and cosine functions to each other. They are based on the fact that the sine and cosine functions are complementary, meaning that they are related to the same angle but in different ways.
Supplementary angles are two angles whose measures add up to 180 degrees. In trigonometry, there are some identities that relate the trigonometric functions of supplementary angles.
sin(180-x) = -sin(x)
cos(180-x) = -cos(x)
tan(180-x) = -tan(x)
Triangle identities are trigonometric identities that relate the sides of a right triangle to the angles. They are based on the Pythagorean theorem and the definitions of the sine, cosine, and tangent functions.
The two most commonly used Pythagorean identities are
sin^2(x) + cos^2(x) = 1
1 + tan^2(x) = sec^2(x)
The two most commonly used co-function identities are:
sin(90-x) = cos(x)
cos(90-x) = sin(x)
Trigonometric identities are used in a variety of ways in mathematics and physics. Some of the main uses include simplifying expressions, solving equations, finding missing sides and angles in triangles, graphing, geometry, physics and so on.
Quotient identities are trigonometric identities that express the ratio of sine and cosine functions in terms of tangent and cotangent functions.