Question

cos theta/1-sin theta + cot theta/1+Sin theta = tan theta + sec theta * cosec theta​

Answer 1

Sinθ/(1 - cosθ)  + Tanθ/(1 + cosθ)  = Secθ.Cosecθ  + Cotθ

Step-by-step explanation:

Sin theta/1-cos theta + tan theta / 1+cos theta = sec theta.cosec theta + cot theta

Sinθ/(1 - cosθ)  + Tanθ/(1 + cosθ)  = Secθ.Cosecθ  + Cotθ

LHS = Sinθ/(1 - cosθ)  + Tanθ/(1 + cosθ)

= (sinθ(1 + cosθ)  + Tanθ(1-Cosθ))/(1 - Cos²θ)

= (sinθ(1 + cosθ) + (Tanθ -  Sinθ)) /Sin²θ

= ( 1 + cosθ  + 1/Cosθ - 1)/Sinθ

= (cosθ  + 1/Cosθ)/Sinθ

= 1/CosθSinθ   + cosθ/Sinθ

= Secθ.Cosecθ + Cotθ

= RHS

QED

Proved

hope it works

Answer 2

$\frac{cos \theta}{1 - sin \theta} + \frac{cot \theta}{1 + sin \theta} \\ \\ = \frac{cos \theta + cos \theta sin \theta + cot \theta - cot \theta sin \theta}{1 - {sin}^{2} \theta} \\ \\ = \frac{cos \theta sin \theta + cot \theta}{ {cos}^{2} \theta } \\ \\ = \frac{sin \theta + \frac{1}{sin \theta} }{cos \theta} \\ \\ = tan \theta + sec \theta cosec \theta$