Question

prove that 1 + sin theta /cos theta + cos theta / 1 + sin theta =
2 sectheta​

Answer 1

Given:

$\dfrac{1+\sin \theta}{\cos \theta}+\dfrac{\cos \theta}{1+\sin \theta}=2\sec \theta$

To prove:

The given statement.

Proof:

We have,

$\dfrac{1+\sin \theta}{\cos \theta}+\dfrac{\cos \theta}{1+\sin \theta}=2\sec \theta$

Taking LHS, we get

$\text{LHS}=\dfrac{1+\sin \theta}{\cos \theta}+\dfrac{\cos \theta}{1+\sin \theta}$

Taking LCM, we get

$\text{LHS}=\dfrac{(1+\sin \theta)^2+cos^2\theta }{\cos \theta(1+\sin \theta)}$

$\text{LHS}=\dfrac{(1)^2+2\sin \theta +\sin^2 \theta+cos^2\theta }{\cos \theta(1+\sin \theta)}$    $[\because (a+b)^2=a^2+2ab+b^2]$

$\text{LHS}=\dfrac{1+2\sin \theta +1}{\cos \theta(1+\sin \theta)}$    $[\because \sin^2 \theta+cos^2\theta =1]$

$\text{LHS}=\dfrac{2+2\sin \theta}{\cos \theta(1+\sin \theta)}$

$\text{LHS}=\dfrac{2(1+\sin \theta)}{\cos \theta(1+\sin \theta)}$

$\text{LHS}=\dfrac{2}{\cos \theta}$

$\text{LHS}=2\sec \theta$         $[\because \sec\theta=\dfrac{1}{\cos \theta}]$

$\text{LHS}=\text{RHS}$

Hence proved.