Question

How to prove that : `1-cos^4A/sin^4A= 1 + 2cot^2A`?

Answer 1

This identity is FALSE

Explanation:

Putting on a common denominator, we get:

`(sin^4A - cos^4A)/sin^4A = 1 + 2cot^2A`

`((sin^2A + cos^2A)(sin^2A - cos^2A))/sin^4A = 1 + 2cot^2A`

`(sin^2A - cos^2A)/sin^4A = 1+ 2cos^2A/sin^2A`

`(sin^2A - (1 - sin^2A))/sin^4A = 1 + 2(cos^2A/sin^2A)`

`(2sin^2A - 1)/sin^4A =(sin^2A + 2cos^2a)/sin^2A`

#(2sin^2A - 1)/sin^4A = (sin^2A + 2(1- sin^2A))/sin^2A

`(2sin^2A - 1)/sin^4A = (2 - sin^2A)/sin^2A`

`2/sin^2A - 1/sin^4A = 2/sin^2A - 1`

And since `1/sin^4A != 1`, this identity is false.

Hopefully this helps!