Question

Prove the identity.(csc²x-1) sin²x = cos²x

Answer 1

Explanation:

We use the 2 identities csc²x = 1/ sin²x and (1 - sin²x) = cos²x:-

(csc²x-1) sin²x

csc²x = 1/ sin²x so

= (1/ sin²x - 1) * sin²x

= [(1 - sin²x) / sin²x] * sin²x

(1 - sin²x) = cos²x so

= (1 - sin²x)

= cos²x

Answer 2

First, remember the following equivalences:

`cscx=(1)/(sinx)`

and:

`cos²x+sin²x=1`

then, given the trigonometric equation, if we use the distributive property and the first equivalence on the left side, we get:

`\begin{gathered} (csc²x-1)sin²x=((1)/(sin²x)-1)sin²x \\ =(sin²x)/(sin²x)-sin²x=1-sin²x \end{gathered}`

then, by the second equivalence:

`\begin{gathered} 1-sin²x=cos²x \\ \Rightarrow cos²x=cos²x \end{gathered}`

therefore, (csc²x-1)=cos²x