Question

please solve this question by using a pen and a paper.​

Answer 1

$\underline{\underline{\bold{Question:}}}$

$\bold{Prove\:\:that:}\\\\\\\tt{sin\:\theta(1+tan\:\theta)+cos\:\theta(1+cot\:\theta)=sec\;\theta+cosec\:\theta.}$

$\tt{Solution:}\\\\\\\underline{\textbf{Solving L.H.S}}\\\\\\\tt{=sin\:\theta(1+tan\:\theta)+cos\:\theta(1+cot\:\theta)}$

$\tt{=sin\:\theta\left(1+\dfrac{sin\:\theta}{cos\:\theta}\right)+cos\theta\left(1+\dfrac{cos\theta}{sin\theta}\right)}\\\\\\\tt{=sin\theta\left(\dfrac{cos\theta+sin\theta}{cos\theta}\right)+cos\theta\left(\dfrac{sin\theta+cos\theta}{sin\theta}\right)}\\\\\\\tt{=\left(\dfrac{sin\theta+cos\theta}{1}\right)\left(\dfrac{sin\theta}{cos\theta}+\dfrac{cos\theta}{sin\theta}\right)}\\\\\\\tt{=\left(\dfrac{sin\theta+cos\theta}{1}\right)\left(\dfrac{sin^2\theta+cos^2\theta}{cos\theta\:sin\theta}\right)}$

$\tt{=\left(\dfrac{sin\theta+cos\theta}{1}\right)\left(\dfrac{1}{sin\theta\:cos\theta}\right)}\\\\\\\tt{=\dfrac{sin\theta+cos\theta}{sin\theta\:cos\theta}}\\\\\\\tt{=\dfrac{sin\theta}{sin\theta\:cos\theta}+\dfrac{cos\theta}{sin\theta\:cos\theta}}\\\\\\\tt{=sec\theta+cosec\theta.}\\\\\\\mathfrak{Hence\:Proved.}$