Question

What is the value of (1/1+cot^2 theta + 1/1+tan^2theta)
?​

Answer 1

$\sf\huge\bold{\underline{\underline{{Solution}}}}$

$\sf \: \dfrac{1}{1 + {cot}^{2} \theta } + \dfrac{1}{1 + {tan}^{2} \theta}$

We know that

Cosec²θ-cot²θ=1

and sec²θ-tan²θ=1

So, 1+cot²θ=cosec²θ ----(1)

sec²θ=1+tan²θ------(2)

from Equation 1 and 2

$\sf \longmapsto \dfrac{1}{ {cosec}^{2} \theta } + \dfrac{1}{{sec}^{2} \theta}$

We know inverse of cosec is sin and inverse of sec is tan

$\sf \large\boxed{ \dfrac{1}{cosec \theta} = sin \theta}$

$\sf \large\boxed{\dfrac{1}{sec \theta } = cos\theta}$

$\sf \longmapsto \: {sin}^{2} \theta + {cos}^{2} \theta$

$\sf \bold{ = 1}$