Cofunction Identities
Cofunction identities in trigonometry give the relationship between the different trigonometric functions and their complementary angles. Let us recall the meaning of complementary angles. Two angles are said to be complementary angles if their sum is equal to π/2 radians or 90°. Cofunction identities are trigonometric identities that show the relationship between trigonometric ratios pairwise (sine and cosine, tangent and cotangent, secant and cosecant). We use the angle sum property of a triangle to derive the six cofunction identities.
In this article, we will derive the cofunction identities and verify them using the sum and difference formulas of trigonometric functions. We will also solve various examples to understand the usage of these cofunction identities to solve various math problems involving trigonometric functions.
What are Cofunction Identities?
Cofunction identities are trigonometric identities that show a relationship between complementary angles and trigonometric functions. We have six such identities that can be derived using a rightangled triangle, the angle sum property of a triangle, and the trigonometric ratios formulas. The cofunction identities give a relationship between trigonometric functions sine and cosine, tangent and cotangent, and secant and cosecant. These functions are referred to as cofunctions of each other. We can also derive these identities using the sum and difference formulas if trigonometric as well. Alternatively, we can use the sum and difference formulas to verify the cofunction identities.
Cofunction Identities Formula
Cofunction identities give a relationship between trigonometric functions pairwise and their complementary angles as below:
 Sine function and cosine function
 Tangent function and cotangent function
 Secant Function and Cosecant Function
Two angles are said to be complementary if their sum is 90 degrees. We can write the cofunction identities in terms of radians and degrees as these are the two units of angle measurement. The six cofunction identities are given in the table below in radians and degrees:
Cofunction Identities in Radians  Cofunction Identities in Degrees 

sin (π/2  θ) = cos θ  sin (90°  θ) = cos θ 
cos (π/2  θ) = sin θ  cos (90°  θ) = sin θ 
tan (π/2  θ) = cot θ  tan (90°  θ) = cot θ 
cot (π/2  θ) = tan θ  cot (90°  θ) = tan θ 
sec (π/2  θ) = cosec θ  sec (90°  θ) = cosec θ 
csc (π/2  θ) = sec θ  csc (90°  θ) = sec θ 
Let us derive these cofunction identities in the next section.
Cofunction Identities Proof
Now that we have discussed the cofunction identities in the previous section, let us now derive them using the right angle triangle. Consider a rightangled triangle ABC right angled at B. Assume angle C = θ, then using the angle sum property of a triangle we have,
∠A + ∠B + ∠C = 180°
⇒ ∠A + 90° + ∠C = 180°  [Because angle B is a right angle]
⇒ ∠A + ∠C = 180°  90°
⇒ ∠A + θ = 90°
⇒ ∠A = 90°  θ
Therefore, we have the three angles of the triangle ABC as ∠A = 90°  θ, ∠B = 90° and ∠C = θ. Now, let us recall the formulas of trigonometric formulas below:
 sin x = Opposite Side / Hypotenuse
 cos x = Adjacent Side / Hypotenuse
 tan x = Opposite Side / Adjacent Side
 cot x = Adjacent Side / Opposite Side
 sec x = Hypotenuse / Opposite Side
 csc x = Hypotenuse / Adjancent Side
Now, using the above formulas, we can determine the cofunction identities for triangle ABC.
 cos θ = BC / AC = sin (90°  θ)
 sin θ = AB / AC = cos (90°  θ)
 tan θ = AB / BC = cot (90°  θ)
 cot θ = BC / AB = tan (90°  θ)
 sec θ = AC / BC = csc (90°  θ)
 csc θ = AC / AB = sec (90°  θ)
Hence, we have derived the cofunction identities. To get these identities in radians, we can simply replace 90° with π/2 and get the identities as:
 cos θ = BC / AC = sin (π/2  θ)
 sin θ = AB / AC = cos (π/2  θ)
 tan θ = AB / BC = cot (π/2  θ)
 cot θ = BC / AB = tan (π/2  θ)
 sec θ = AC / BC = csc (π/2  θ)
 csc θ = AC / AB = sec (π/2  θ)
Verification of Cofunction Identities
Now that we have proved the cofunction identities, let us verify them using the sum and difference formulas of trigonometry. We will use the following formulas to verify the identities:
 sin(A  B) = sinA cosB  cosA sinB
 cos(A  B) = cosA cosB + sinA sinB
 tan A = sin A / cos A
Expand sin (π/2  θ), cos (π/2  θ), and tan (π/2  θ) using the above formulas.
 sin (π/2  θ) = sin(π/2) cosθ  cos(π/2) sinθ
= 1 × cos θ  0 × sin θ  [Because sin (π/2) = 1 and cos (π/2) = 0]
= cos θ  cos (π/2  θ) = cos(π/2) cosθ + sin(π/2) sinθ
= 0 × cos θ + 1 × sin θ  [Because sin (π/2) = 1 and cos (π/2) = 0]
= sin θ  tan(π/2  θ) = [sin (π/2  θ)] / [cos (π/2  θ)]
= cos θ / sin θ
= cot θ
Let us now verify the cofunction identities for sec, csc, and cot using reciprocal identities
 cot (π/2  θ) = 1 / tan (π/2  θ)
= 1 / cot θ
= tan θ  sec (π/2  θ) = 1 / cos (π/2  θ)
= 1 / sin θ
= csc θ  csc (π/2  θ) = 1 / sin (π/2  θ)
= 1 / cos θ
= sec θ
Hence, we have verified all six cofunction identities using trigonometric formulas.
Using Cofunction Identities
Now that we have derived the formulas for the cofunction identities, let us solve a few problems to understand its application.
Example 1: Find the value of acute angle x, if sin x = cos 20°.
Solution: Using cofunction identity, cos (90°  θ) = sin θ, we can write sin x = cos 20° as
sin x = cos 20°
⇒ cos (90°  x) = cos 20°
⇒ 90°  x = 20°
⇒ x = 90°  20°
⇒ x = 70°
Answer: Value of x is 70° if sin x = cos 20°.
Example 2: Evaluate the value of x, if sec (5x) = csc (x + 18°), where 5x is an acute angle.
Solution: To find the value of x, we will use the cofunction identity csc (90°  θ) = sec θ. We can write
sec (5x) = csc (x + 18°)
⇒ csc (90°  5x) = csc (x + 18°)
⇒ 90°  5x = x + 18°  [Because it is given 5x is acute]
⇒ 5x + x = 90°  18°
⇒ 6x = 72°
⇒ x = 72° / 6
⇒ x = 12°
Answer: Value of x is 12° if sec (5x) = csc (x + 18°), where 5x is an acute angle.
Important Notes on Cofunction Identities
 Cofunction identities show the relationship between trigonometric functions and complementary angles.
 We have main six cofunction identities:
 cos θ = sin (90°  θ)
 sin θ = cos (90°  θ)
 tan θ = cot (90°  θ)
 cot θ = tan (90°  θ)
 sec θ = csc (90°  θ)
 csc θ = sec (90°  θ)
 These identities can be derived using the angle sum property of a right triangle and sum and difference formulas.
☛ Related Topics:
Cofunction Identities Examples

Example 1: Determine the value of sin 150° using cofunction identities.
Solution: To find the value of sin 150°, we will use the formula sin θ = cos (90°  θ). So, we have
sin 150° = cos (90°  150°)
= cos (60°)
= cos (60°)  [Because cos (x) = cos x for all x.]
= 1/2  [Because cos 60° = 1/2]
Answer: sin 150° = 1/2

Example 2: Find the value of tan 30° + cot 150° using cofunction identities.
Solution: To find the value tan 30° + cot 150°, we will use first the values of tan 30° and cot 150°, separately.
tan 30° = 1/√3
cot 150° = 1 / tan 150°  [Because tan and cot are reciprocals of each other.]
= 1 / tan (90° + 60°)
= 1 / tan (90°  (60°))
= 1 / cot (60°)  [Using cofunction identity cot θ = tan (90°  θ)]
=  1 / cot 60°
= 1 / √3
So, we have tan 300° + cot 150° = 1/√3  1/√3 = 0.
Answer: tan 300° + cot 150° = 0

Example 3: Find the value of θ if tan θ = cot (θ/2 + π/12) using cofunction identities.
Solution: To find the value of θ, we will use the formula tan θ = cot (π/2  θ). So, we have
tan θ = cot (θ/2 + π/12)
⇒ cot (π/2  θ) = cot (θ/2 + π/12)
⇒ π/2  θ = θ/2 + π/12
⇒ θ + θ/2 = π/2  π/12
⇒ 3θ/2 = 6π/12  π/2
⇒ θ = 5π/12 × 2/3
= 5π/18
Answer: θ = 5π/18
FAQs on Cofunction Identities
What are Cofunction Identities in Trigonometry?
Cofunction identities in trigonometry are formulas that show the relationship between trigonometric functions and their complementary angles pairwise  (sine and cosine, tangent and cotangent, secant and cosecant). We have mainly six cofunction identities that are used to solve various problems in trigonometry.
What are the Main Six Cofunction Identities?
The six main cofunction identities are:
 cos θ = sin (90°  θ)
 sin θ = cos (90°  θ)
 tan θ = cot (90°  θ)
 cot θ = tan (90°  θ)
 sec θ = csc (90°  θ)
 csc θ = sec (90°  θ)
We can write these identities using the measure of radians also as given below:
 cos θ = sin (π/2  θ)
 sin θ = cos (π/2  θ)
 tan θ = cot (π/2  θ)
 cot θ = tan (π/2  θ)
 sec θ = csc (π/2  θ)
 csc θ = sec (π/2  θ)
How Do You Find Cofunction Identities?
We can derive the formulas for the six cofunction identities using a rightangled triangle and the angle sum property of a triangle. We can also prove these identities using the sum and difference formulas and reciprocal identities in trigonometry.
What are Cofunction Identities For Tangent and Cotangent?
The cofunction identities for tangent and cotangent are given below:
 tan θ = cot (π/2  θ)
 cot θ = tan (π/2  θ)
We can also write these formulas in terms of degrees also as:
 tan θ = cot (90°  θ)
 cot θ = tan (90°  θ)
Why are Cofunction Identities True for all Right Triangles?
We say that two functions are cofunctions of each other if their angles are complementary, that is, the sum of their angles is π/2 rad. In an arbitrary right triangle, since one angle is π/2 rad, the sum of the other two angles is always π/2 using the nagle sum property. So, the cofunction identities are true for all right triangles and they can be easily derived using a right triangle and applying trigonometric ratios formulas to it.
When to Use Cofunction Identities?
We can use cofunction identities to simplify various complex trigonometric problems. They are used when the angles involved are complementary, that is, their sum is 90 degrees. Cofunction identities can be used to find values of trigonometric ratios with angles more than 90 degrees to simplify them.
visual curriculum