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