Question

Use the fundamental identities to simplify the expression. There is more than one correct form of the answer.

Answer 1

From the fundamental identities, we know that

`csc\varphi=(1)/(sin\varphi)`

By substituting this result into the given expression, we have

`9sin\varphi((1)/(sin\varphi)-sin\varphi)`

Now, by distributing sine of phi into the parentheses, we have

`9((sin\varphi)/(sin\varphi)-sin^2\varphi)`

or equivalently,

`9(1-sin^2\varphi)`

Now, from the Pythagorean identity:

`cos^2\varphi+sin^2\varphi=1`

we can note that

`cos^2\varphi=1-sin^2\varphi`

Then, by substituting this result into our last result from above, we obtain

`9(1-sin^2\varphi)=9cos^2\varphi`

Therefore, the answer is:

`9cos^2\varphi`