Question

How do you verify the identity `(2+cscthetasectheta)/(cscthetasectheta)=(sintheta+costheta)^2`?

Answer 1

You will need the following identities to prove this identity.

•`cscbeta = 1/sinbeta`

•`secbeta = 1/cosbeta`

•`sin^2beta + cos^2beta = 1`

Simplify the left-hand side and expand the right-hand side.

`(2 + 1/sintheta xx 1/costheta)/(1/sintheta xx 1/costheta) = sin^2theta+ 2sinthetacostheta + cos^2theta`

`(2 + 1/(sinthetacostheta))/(1/(sinthetacostheta)) = 1 + 2sinthetacostheta`

`((2sinthetacostheta + 1)/(sinthetacostheta))/(1/(sinthetacostheta)) = 1 + 2sinthetacostheta`

`(2sinthetacostheta+ 1)/(sin thetacostheta) xx (sinthetacostheta)/1 = 1+ 2sinthetacostheta`

`2sinthetacostheta + 1 = 1 + 2sinthetacostheta`

Identity proved!!

Hopefully this helps!