Question

Prove that sin theta divided by 1 cos theta+tan theta/1+cos theta=cot theta+chick+sec theta*cosec theta

Answer 1

writing tanθ assinθcosθ and cotθ as cosθsinθ, we get

sinθcosθ1−cosθsinθ+cosθsinθ1−sinθcosθ

=sin2θcosθ⋅(sinθ−cosθ)+cos2θsinθ⋅(cosθ−sinθ) (how?)

=sin2θcosθ⋅(sinθ−cosθ)−cos2θsinθ⋅(sinθ−cosθ)

=1(sinθ−cosθ)(sin2θcosθ−cos2θsinθ)

=1(sinθ−cosθ)(sin3θ−cos3θsinθ⋅cosθ)

=sinθ−cosθsinθ−cosθ(sin2θ+sinθ⋅cosθ+cos2θ)sinθ⋅cosθ(how?)

=1⋅1+sinθ⋅cosθsinθ⋅cosθ (why?)

which is

1+secθ⋅cscθ