Question

Prove (SinØ+cosø)(tanø+cotø)=secø+cosecø

Answer 1

To prove----->

(Sinφ + Cosφ) (tanφ + Cotφ) = Secφ + Cosecφ

To prove-----> LHS

= ( Sinφ + Cosφ ) ( tanφ + Cotφ )

We know that,

tanφ = Sinφ / Cosφ , Cotφ = Cosφ / Sinφ , applying it we get,

= ( Sinφ + Cosφ ) ( Sinφ/Cosφ + Cosφ/Sinφ )

= ( Sinφ + Cosφ ) { ( Sin²φ + Cos²φ ) / Sinφ Cosφ }

We know that,

Sin²θ + Cos²θ = 1 , applying it we get,

= ( Sinφ + Cosφ ) ( 1 / Sinφ Cosφ )

= ( Sinφ + Cosφ ) / Sinφ Cosφ

= Sinφ / Sinφ Cosφ + Cosφ / Sinφ Cosφ

= 1 / Cosφ + 1 / Sinφ

We know that,

Secθ = 1 / Cosθ , Cosecθ = 1 / Sinθ , applying it , we get,

= Secφ + Cosecφ = RHS