Trigonometric identities

Trigonometric identities

For all real values of θ , we have the following three identities.

(i) sin ² θ + cos² θ=1

(ii) 1 + tan² θ = sec² θ

(iii) 1 + cot² θ = cosec² θ

These identities are termed as three fundamental identities of trigonometry. We will now prove them as follows.

These identities can also be rewritten as follows.

Note

Though the above identities are true for any angle θ, we will consider the six trigonometric ratios only for 0° < θ < 90°

Example 6.1

Prove that tan ² θ − sin² θ = tan ² θ sin² θ

Solution

tan ² θ - sin² θ = tan² θ − . cos² θ

 = tan ² θ(1 − cos ² θ) = tan ² θ sin² θ

Example 6.2

Prove that 

Solution

Example 6.3

Prove that 1 +  = cosec θ

Solution

Example 6.4

Prove that sec θ − cos θ = tan θ sin θ

Solution

Example 6.5 Prove that  = cosec θ + cot θ

Solution

Example 6.6

Prove that  = cot θ

Solution

Example 6.7

Prove that sin ² A cos² B + cos ² A sin² B + cos ² A cos² B + sin ² A sin² B = 1

Solution

sin ² A cos² B + cos ² A sin² B + cos ² A cos² B + sin ² A sin² B

 = sin ² A cos² B + sin ² A sin² B + cos ² A sin² B + cos ² A cos² B 

 = sin² A(cos² B + sin² B ) + cos² A(sin² B + cos ² B)

 = sin ² A(1) + cos² A(1)              (since  sin² B + cos ² B = 1)

 = sin ² A + cos² A = 1

Example 6.8

If cos θ + sin θ = √2 cos θ, then prove that cos θ − sin θ =  √2sin θ

Solution

Now, cos θ + sin θ = √2 cos θ

Squaring both sides,

(cos θ + sin θ)² = (√2 cos θ)²

cos ² θ + sin² θ + 2 sin θ cos θ = 2 cos² θ

2 cos² θ - cos² θ - sin² θ = 2 sin θ cos θ

cos ² θ - sin² θ = 2 sin θ cos θ

(cos θ + sin θ)(cos θ − sin θ) =2 sin θ cos θ

Therefore cos θ − sin θ = √2sin θ

Example 6.9 

Prove that (cosec θ − sin θ)(sec θ − cos θ)(tan θ + cot θ) =1

Solution

(cosec θ − sin θ)(sec θ − cos θ)(tan θ + cot θ)

Example 6.10

Prove that 

Solution 

Example 6.11 

If cosec θ +cot θ = P , then prove that cos θ = 

Solution 

Given cosec θ +cot θ = P   ...(1)

cosec² θ - cot² θ =1 (identity)

cosec θ - cot θ = 1 / (cosec θ + cot θ)

Example 6.12

Prove that tan²A − tan² B =  

Solution

Example 6.13

Solution

Example 6.14 Prove that 

Solution

Example 6.15

Solution

Example 6.16

Prove that  = sin ² A cos² A

Solution

Example 6.17

Solution