Question

prove the identity (sin theta+cosec theta)²+(cos theta+sec theta)²=7+tan²theta+cot²theta​

Answer 1

Answer:

$\Large{\underline{\underline{\bf{Given:}}}}$

• (Sinθ+cosecθ)²+(cosθ+secθ)² = 7+tan²θ+cot²θ

$\Large{\underline{\underline{\bf{To\:Prove:}}}}$

• LHS=RHS

$\Large{\underline{\underline{\bf{Proof:}}}}$

↬ Consider the LHS of the equation

↬ LHS=

    (sinθ+cosecθ)²+(cosθ+secθ)²

 =  sin²θ+cosec²θ+2sinθcosecθ + cos²θ+sec²θ+2cosθsecθ

 = sin²θ+cos²θ+2sinθcosecθ+cosec²θ+sec²θ+2cosθsecθ

 =  1+2×1+(1+cot²θ)+(1+tan²θ)+2×1

 = 1+2+1+1+2+cot²θ+tan²θ

 = 7+tan²θ+cot²θ

 = RHS

↬ Hence proved

$\Large{\underline{\underline{\bf{Identities\:used:}}}}$

↬ (a+b)²= a²+b²+2ab

↬ sin²θ+cos²θ = 1

↬ sinθcosecθ = 1

↬ cosec²θ = 1+cot²θ

↬ sec²θ =1+tan²θ

↬ cosθsecθ = 1