# 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