Question

How do you verify `(csc x+sec x)/(sin x+ cos x)=cot x+tan x`?

Answer 1

For an identity like this, we have to be clear with the following identities.

Explanation:

The reciprocal identities

`csctheta = 1/sintheta`

`sectheta = 1/costheta`

`cottheta = 1/tantheta`

The quotient identities:

`tantheta = sin theta/costheta`

`cottheta = costheta/sintheta`

Applying all these identities, on both sides, we get:

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

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

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

`1/(cosxsinx)= (cos^2x + sin^2x)/(sinxcosx)`

Applying the pythagorean identity `sin^2x + cos^2x = 1` on the right side, we get:

`1/(cosxsinx) = 1/(sinxcosx)`

Hopefully this helps!