Question

1/1+sin^2theta+1/1+cos^2theta+1/1+sec^2theta+1/1+cosec^2theta=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

Answer with Step-by-step explanation:

To prove: $\dfrac{1}{1+sin^2\theta} + \dfrac{1}{1+cos^2 \theta} +\dfrac{1}{1+ sec^2 \theta} + \dfrac{1}{1+cosec ^2 \theta}=2$

$L.H.S = \dfrac{1}{1+sin^2\theta} + \dfrac{1}{1+cos^2 \theta} +\dfrac{1}{1+ sec^2 \theta} + \dfrac{1}{1+cosec ^2 \theta}\\\\\\=\dfrac{1}{1+sin^2\theta} + \dfrac{1}{1+cos^2 \theta} +\dfrac{1}{1+ \dfrac{1}{cos^2 \theta}} + \dfrac{1}{1+\dfrac{1}{sin^2\theta} }\\\\\ = \dfrac{1}{1+sin^2\theta} + \dfrac{1}{1+cos^2 \theta} +\dfrac{cos^2 \theta}{ cos^2 \theta+1} + \dfrac{sin^2\theta}{sin ^2 \theta+1}$

$= \dfrac{1}{1+sin^2\theta}+ \dfrac{sin^2\theta}{sin ^2 \theta+1} +\dfrac{cos^2 \theta}{ cos^2 \theta+1} +\dfrac{1}{1+cos^2 \theta}\\\\\\=\dfrac{1+sin^2\theta}{sin ^2 \theta+1}+\dfrac{1+cos^2 \theta}{ cos^2 \theta+1}\\\\=1+1=2=L.H.S.$

Hence Proved