Let us see trigonometric identities
- sin² θ + cos² θ = 1
- sin² θ = 1 - cos² θ
- cos² θ = 1 - sin² θ
- Sec² θ - tan² θ = 1
- Sec² θ = 1 + tan² θ
- tan² θ = Sec² θ - 1
- Cosec² θ - cot² θ = 1
- Cosec² θ = 1 + cot² θ
- cot² θ = Cosec² θ - 1
These identities are applied in both ways ,left to right or right to left.So we have to memories all the identities.
We are going to see the example problems based on the above identities.
Example 1:
Prove that sin⁴ θ + cos⁴ θ = 1 - 2 sin² θ cos² θ
Solution:
L.H.S
= sin⁴ θ + cos⁴ θ
= (sin² θ)² + (cos² θ)²
a² + b² = (a+b)² - 2ab
= (sin² θ + cos² θ)² - 2 (sin² θ) (cos² θ)
= (1)² - 2 sin² θ cos² θ
= 1 - 2 sin² θ cos² θ
R.H.S
Hence it is proved