Question

How do you prove `(sinx+cosx)/(secx+cscx)=sinx/secx`?

Answer 1

See below.

Explanation:

Apply the following identities:

•`sectheta = 1/costheta`

•`csctheta = 1/sintheta`

Now, simplify both sides using the given identities:

`(sinx + cosx)/(1/cosx + 1/sinx) = sinx/(1/cosx)`

`(sinx + cosx)/((sinx + cosx)/(sinxcosx)) = sinxcosx`

`sinx + cosx xx (sinxcosx)/(sinx + cosx) = sinxcosx`

`cancel(sinx + cosx) xx (sinxcosx)/(cancel(sinx+cosx)) = sinxcosx`

`sinxcosx = sinxcosx`

Identity proved!!

Hopefully this helps!