Question

Evalute (1+tan theta+sec theta)(1+cot theta-cosec theta)

Answer 1

Notice,

(1+cotθ−cosecθ)(1+tanθ+secθ)=(1+cosθsinθ−1sinθ)(1+sinθcosθ+1cosθ)

=(sinθ+cosθ−1sinθ)(sinθ+cosθ+1cosθ)

=(sinθ+cosθ)2−12sinθcosθ

=sin2θ+cos2θ+2sinθcosθ−1sinθcosθ

=1+2sinθcosθ−1sinθcosθ

=2sinθcosθsinθcosθ

=2

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