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

2 Answers
Mar 25, 2018

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.

Mar 25, 2018

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