prove that (sec²+tan²)(csc²+cot²)=1+2sec²csc²

Question

the solution of the above question please

Answer 1

Let us see trigonometric identities

1. sin² θ  + cos² θ = 1
2. sin² θ  = 1 - cos² θ
3. cos² θ = 1 - sin² θ
4. Sec² θ - tan² θ = 1
5. Sec² θ  = 1 +  tan² θ
6. tan² θ  =  Sec² θ - 1
7. Cosec² θ - cot² θ = 1
8. Cosec² θ = 1 + cot² θ
9. 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