Proving Trig Identities

[f1player]

Prove: (cot a + cosec a)^2 = (1 + cos a)/(1 - cos a)

I get stuck halfway through

I did this

LHE: ((cos a)/(sin a) + 1/(sin a))^2

= ((cos a + 1)^2/(sin a))^2

= (cos a + 1)(cos a + 1)/(sin^2 a)

= (cos^2 a + 2 cosa + 1)/(sin^2a)

And here I get stuck

What is the next step??

 

[soroban]

Hello, f1player!

You reached the right point . . . but went off.

Prove: \(\displaystyle \,(\cot a\,+\,\csc a)^2 \:=\:\L\frac{1\,+\,\cos a}{1\,-\,\cos a}\)

I get stuck halfway through

I did this

LHS: \(\displaystyle \L\,\left(\frac{\cos a}{\sin a}\,+\,\frac{1}{\sin a}\right)^2\)

\(\displaystyle \L\;\;\;=\;\frac{(\cos a\,+\,1)^2}{\sin^2a}\) . . . here!

\(\displaystyle \L\;\;\;\;=\;\frac{(1\,+\,cos a)^2}{1\,-\,\cos^2a}\:=\:\frac{(1\,+\,\cos a)^2}{(1\,-\,\cos a)(1\,+\,\cos a)}\;=\;\frac{1\,+\,cos a}{1\,-\,\cos a}\)

 

[f1player]

Thanks, I didnt see that one!