Question

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

Answer 1

Given:

`(1+\sin \theta)/(\cos \theta)+(\cos \theta)/(1+\sin \theta)=2\sec \theta`

To prove:

The given statement.

Proof:

We have,

`(1+\sin \theta)/(\cos \theta)+(\cos \theta)/(1+\sin \theta)=2\sec \theta`

Taking LHS, we get

`\text{LHS}=(1+\sin \theta)/(\cos \theta)+(\cos \theta)/(1+\sin \theta)`

Taking LCM, we get

`\text{LHS}=((1+\sin \theta)^2+cos^2\theta )/(\cos \theta(1+\sin \theta))`

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

`\text{LHS}=(2+2\sin \theta)/(\cos \theta(1+\sin \theta))`

`\text{LHS}=(2(1+\sin \theta))/(\cos \theta(1+\sin \theta))`

`\text{LHS}=(2)/(\cos \theta)`

`\text{LHS}=2\sec \theta`
`[\because \sec\theta=(1)/(\cos \theta)]`

`\text{LHS}=\text{RHS}`

Hence proved.