Question

Prove that:

SinA/cotA+cosecA =2+(sinA/cotA - cosecA)

Answer 1

Take L.H.S
=>
SinA/cotA+cosecA

=sinA/(cosA/sinA+1/sinA)

=sinA/{(cosA+1)/sinA}

=sin²A/(1+cosA)

=(1-cos²A)/(1+cosA)

=(1+cosA)(1-cosA)/(1+cosA)

=1-cosA

Now take R.H.S
2+sinA/cotA-cosecA

=2+sinA/(cosA/sinA-1/sinA)

=2+sinA/{(cosA-1)/sinA}

=2+sin²A/(cosA-1)

=2+(1-cos²A)/{-(1-cosA)}

=2-(1+cosA)(1-cosA)/(1-cosA)

=2-(1+cosA)

=2-1-cosA

=1-cosA
∴, LHS=RHS (Proved)
I hope you understood the answer :)