Question

prove that sinteta(1+tanteta) + costeta(1+cotteta) = secteta + cosecteta​

Answer 1

Answer:

Secθ + Cosecθ

Step-by-step explanation:

To prove--->

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Sinθ(1+ tanθ)+Cosθ(1+Cotθ)=Secθ +

Cosecθ

Proof--->

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LHS=Sinθ(1+tanθ)+Cosθ(1+Cotθ)

=Sinθ(1+ Sinθ/Cosθ)+Cosθ(1+Cosθ /

Sinθ)

=Sinθ(Cosθ +Sinθ/Cosθ) + Cosθ(Sinθ +Cosθ /Sinθ )

=(Cosθ + Sinθ) (Sinθ/Cosθ + Cosθ/Sinθ)

=(Cosθ+Sinθ) (Sin²θ + Cos²θ/Sinθ Cosθ)

We know that

Sin²θ + Cos²θ =1 , applying it here

=(Cosθ+Sinθ) ( 1/Sinθ Cosθ)

=(Cosθ/Sinθ Cosθ + Sinθ / Sinθ Cosθ)

= 1 / Sinθ + 1/ Cosθ

We know that

1/Sinθ =Cosecθ and 1/Cosθ = Secθ

applying these identities here

= Cosecθ + Secθ =RHS