prove sin theta(1+tan theta)+cos theta(1+cot theta)=sec theta+cosec theta
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Answer:Sinθ(1 + Tanθ) + Cosθ(1 + Cotθ) = Secθ + Cosecθ
Step-by-step explanation:
Sin theta( 1+tan theta) + cos theta(1+ cot theta) = sec theta + cosec theta
Sinθ(1 + Tanθ) + Cosθ(1 + Cotθ) = Secθ + Cosecθ
LHS = Sinθ(1 + Tanθ) + Cosθ(1 + Cotθ)
Using Tanθ = Sinθ/Cosθ & Cotθ = Cosθ/Sinθ
= Sinθ(1 + Sinθ/Cosθ) + Cosθ(1 + Cosθ/Sinθ)
= (Sinθ/Cosθ)(Cosθ + Sinθ) + (Cosθ/Sinθ)(Sinθ + Cosθ)
= (Cosθ + Sinθ) (Sinθ/Cosθ + Cosθ/Sinθ)
= (Cosθ + Sinθ)((Sin²θ + Cos²θ)/CosθSinθ)
Using Sin²θ + Cos²θ = 1
= (Cosθ + Sinθ)/CosθSinθ
= Cosθ/CosθSinθ + Sinθ/CosθSinθ
= 1/Sinθ + 1/Cosθ
= Cosecθ + Secθ
= Secθ + Cosecθ
= RHS
QED
Proved
Sinθ(1 + Tanθ) + Cosθ(1 + Cotθ) = Secθ + Cosecθ
Step-by-step explanation:
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