Question

Answer the question.....​

Answer 1

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

$\bold{Simplify:}\\\\\\\tt{cos\theta\sqrt{1+tan^2\theta}+sin\theta\sqrt{1+cot^2\theta.}}$

$\mathfrak{Answer:}$

= 2 .

$\mathfrak{Step-by-Step\:Explanation:}$

$\tt{=cos\theta\sqrt{1+tan^2\theta}+sin\theta\sqrt{1+cot^2\theta}}\\\\\\\boxed{\bold{1+tan^2\theta=sec^2\theta\quad\&\quad 1+cot^2\theta=cosec^2\theta}}\\\\\\\tt{=cos\theta\sqrt{sec^2\theta}\:+\:sin\theta\sqrt{cosec^2\theta}}\\\\\\\tt{=cos\theta\:sec\theta\:+\:sin\theta\:cosec\theta}\\\\\\\boxed{\bold{sec\theta=\dfrac{1}{cos\theta}\quad\&\quad\:cosec\theta=\dfrac{1}{sin\theta}}}\\\\\\\tt{=cos\theta\times\dfrac{1}{cos\theta}+sin\theta\times\dfrac{1}{sin\theta}}\\\\\\\tt{=1+1=2.}$