How can I prove this equation is an identity? sin^4w=1-cot^2w + cos^2w (cot^2w)/(csc^2w)
2 Answers
This is not an identity.
Explanation:
The left hand side is
while the right hand side simplifies to
Since
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
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