Question

How to solve this identity? Thank you!

`1/sina + 1/tana = cot(a/2)`

Answer 1

If we know our trigonometric inverses right, we can simplify as follows:
`1/sin(a)=csc(a)`
`1/tan(a)=cot(a)`

`1/sin(a)+1/tan(a)=csc(a)+cot(a)`

This is just the result of the half angle formula for `cot`:
`cot(theta/2)=csc(theta)+cot(theta)`

So we can simplify to:
`=cot(a/2)`, which is what we wanted to prove.

Answer 2

See explanation.

Explanation:

`L=1/sinalpha+1/tanalpha=1/sinalpha+cosalpha/sinalpha=`

`=(1+cosalpha)/sinalpha=(1+cos2*(alpha/2))/sin(2*alpha/2)=`

`(1+2cos^2(alpha/2)-1)/(2sin(alpha/2)cos(alpha/2))=`

`(2cos^2(alpha/2))/(2sin(alpha/2)cos(alpha/2))=`

`(cos^(cancel(2)}(alpha/2))/(sin(alpha/2)cancel(cos(alpha/2)))=`

`(cos(alpha/2))/(sin(alpha/2))=cot(alpha/2)=R`