Question

How can I prove this equation is an identity? `sin^4w=1-cot^2w + cos^2w (cot^2w)/(csc^2w)`

Answer 1

This is not an identity.

Explanation:

The left hand side is

`sin^4w = (1-cos^2w)^2=1-color(red)(2cos^2w)+cos^4w`

while the right hand side simplifies to

`1-cot^2w + cos^2w (cot^2w)/(csc^2w) = 1-cot^2w + cos^2w (cos^2w)/(sin^2wcsc^2w) = 1-color(red)(cot^2w)+cos^4w`

Since `cot^2w ne 2cos^2w` in general, this is not an identity.

Answer 2

it is not an identity

Explanation:

We seek to prove:

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

We can readily disprove the claim using a counter example:

Consider the case `w=pi/6`, then:

`LHS = sin^4 (pi/6)`
`\ \ \ \ \ \ \ \ = (1/2)^4`
`\ \ \ \ \ \ \ \ = 1/16`

And:

`RHS = 1 - cot^2(pi/6) + cos^2(pi/6) cot^2(pi/6)/csc^2(pi/6)`

`\ \ \ \ \ \ \ \ = 1 - (sqrt(3))^2 + (sqrt(3)/2)^2 (sqrt(3))^2/(2)^2`

`\ \ \ \ \ \ \ \ = 1 - 3 + (3/4) 3/4`

`\ \ \ \ \ \ \ \ = -2 + 9/16`

`\ \ \ \ \ \ \ \ = -23/16`

`\ \ \ \ \ \ \ \ != LHS`

Hence this is not an identity