Question

Prove that (sinθ - cosecθ)(cosθ - secθ) = 1/(tanθ + cotθ)

Answer 1

LHS=(1/sinФ - sinФ) (1/cosФ -cosФ)

=((1-sin²Ф)/sinФ) ((1-cos²Ф)/cosФ)

=(cos²Ф/sinФ) (sin²Ф/cosФ)

=cosФsinФ ----1

RHS = 1/(sinФ/cosФ +cosФ/sinФ)

=1/((sin²Ф+cos²Ф)/(sinФcosФ))

=1/(1/cosФ)(1/sinФ)

=cosФsinФ----2

LHS = RHS

 (cosecФ-sinФ)(secФ-cosФ)=1/tanФ+cotФ