Question

(sectheta-cosectheta) (1+tantheta+cottheta)=tantheta sectheta-cottheta cosectheta. prove that​

Answer 1

Answer:

(Secθ -Cosecθ)(1 + Tanθ + Cotθ) = TanθSecθ - CotθCosecθ

Step-by-step explanation:

(Secθ -Cosecθ)(1 + Tanθ + Cotθ) = Tanθ Secθ - CotθCosecθ

LHS

= (Secθ -Cosecθ)(1 + Tanθ + Cotθ)

= (1/Cosθ - 1/Sinθ)(1  + Sinθ/Cosθ + Cosθ/Sinθ)

= {(Sinθ - Cosθ)/(CosθSinθ) }{CosθSinθ  + Sin²θ + Cos²θ)/CosθSinθ) }

= (1 + CosθSinθ)(Sinθ - Cosθ))/(Cos²θSin²θ)

= (Sinθ - Cosθ  + Sin²θCosθ - Cos²θSinθ)/(Cos²θSin²θ)

= (Sinθ(1 - Cos²θ)  -Cosθ(1 - Sin²θ)/(Cos²θSin²θ)

= (SinθSin²θ - CosθCos²θ)/(Cos²θSin²θ)

= Sinθ/Cos²θ  - Cosθ/Sin²θ

= Tanθ/Cosθ - Cotθ/Sinθ

= TanθSecθ - CotθCosecθ

= RHS

QED

Proved

(Secθ -Cosecθ)(1 + Tanθ + Cotθ) = TanθSecθ - CotθCosecθ