Question

sin theta( 1+tan theta) + cos theta(1+ cot theta) = sec theta + cosec theta we have to prove tell me the answer quickly

Answer 1

solution,

Given,

sinθ(1+tanθ)+cosθ(1+cotθ)=secθ+cosecθ

now,

LHS= sinθ(1+tanθ)+cosθ(1+cotθ)

$= sin\theta(1+\frac{sin\theta}{cos\theta})+cos\theta(1+\frac{cos\theta}{sin\theta})\\\\= sin\theta(\frac{cos\theta+sin\theta}{cos\theta})+cos\theta(\frac{sin\theta+cos\theta}{sin\theta})\\\\=(sin\theta+cos\theta)[\frac{sin\theta}{cos\theta}+\frac{cos\theta}{sin\theta}]\\\\= (sin\theta+cos\theta) [tan\theta+cot\theta]\\\\=(sin\theta+cos\theta)[tan\theta+\frac{1}{tan\theta} ]\\\\=(sin\theta+cos\theta)[\frac{tan^2\theta+1}{tan\theta}]\\\\=(sin\theta+cos\theta)[\frac{sec^2\theta}{tan\theta}]$

$= (sin\theta+cos\theta)[sec^2\theta*cot\theta]\\\\=(sin\theta+cos\theta)[\frac{1}{cos^2\theta}*\frac{cos\theta}{sin\theta}]\\\\=\frac{(sin\theta+cos\theta)}{cos\theta*sin\theta}\\\\=\frac{sin\theta}{cos\theta*sin\theta}+\frac{cos\theta}{cos\theta*sin\theta}\\\\=\frac{1}{cos\theta}+\frac{1}{sin\theta}\\\\= sec\theta+cosec\theta$

RHS proved