Question

1/1+sin²theta+1/1+cos²theta+1/1+sec²theta 1/1+cosec²theta=2

Answer 1

$LHS=\frac{1}{1+sin^2\theta}+\frac{1}{1+cos^2\theta}+\frac{1}{1+sec^2\theta}+\frac{1}{1+cosec^2\theta}\\\\\\=\frac{1}{1+sin^2\theta}+\frac{1}{1+cos^2\theta}+\frac{1}{1+\frac{1}{cos^2\theta}}+\frac{1}{1+\frac{1}{sin^2\theta}}\\\\\\=\frac{1}{1+sin^2\theta}+\frac{1}{1+cos^2\theta}+\frac{cos^2\theta}{cos^2\theta+1}+\frac{sin^2\theta}{sin^2\theta+1}\\\\\\=\frac{1}{1+sin^2\theta}+\frac{sin^2\theta}{1+sin^2\theta}+\frac{1}{1+cos^2\theta}+\frac{cos^2\theta}{1+cos^2\theta}\\\\\\=\frac{1+sin^2\theta}{1+sin^2\theta}+\frac{1+cos^2\theta}{1+cos^2\theta}\\\\\\=1+1=2=RHS$

Answer 2

Hey there!

Taking L.H.S,

$LHS=\dfrac{1}{1+sin^2\theta}+\dfrac{1}{1+cos^2\theta}+\dfrac{1}{1+sec^2\theta}+\dfrac{1}{1+cosec^2\theta}\\\\\\=\dfrac{1}{1+sin^2\theta}+\dfrac{1}{1+cos^2\theta}+\dfrac{1}{1+\frac{1}{cos^2\theta}}+\dfrac{1}{1+\dfrac{1}{sin^2\theta}}\\\\\\=\dfrac{1}{1+sin^2\theta}+\frac{1}{1+cos^2\theta}+\dfrac{cos^2\theta}{cos^2\theta+1}+\dfrac{sin^2\theta}{sin^2\theta+1}\\\\\\=\dfrac{1}{1+sin^2\theta}+\dfrac{sin^2\theta}{1+sin^2\theta}+\dfrac{1}{1+cos^2\theta}+\dfrac{cos^2\theta}{1+cos^2\theta}$$\\\\\\=\dfrac{1+sin^2\theta}{1+sin^2\theta}+\dfrac{1+cos^2\theta}{1+cos^2\theta}\\\\\\=1+1\\=2\\=RHS$

Hope It Helps You!