Question

Find the value of theta, if, Cos (theta) / 1 - sin (theta) + cos (theta) / 1 + sin (theta) = 4 ?

Answer 1

`theta=pi/3` or `60^@`

Explanation:

Okay. We've got:

`costheta/(1-sintheta)+costheta/(1+sintheta)=4`

Let's ignore the `RHS` for now.

`costheta/(1-sintheta)+costheta/(1+sintheta)`

`(costheta(1+sintheta)+costheta(1-sintheta))/((1-sintheta)(1+sintheta))`

`(costheta((1-sintheta)+(1+sintheta)))/(1-sin^2theta)`

`(costheta(1-sintheta+1+sintheta))/(1-sin^2theta)`

`(2costheta)/(1-sin^2theta)`

According to the Pythagorean Identity,

`sin^2theta+cos^2theta=1`. So:

`cos^2theta=1-sin^2theta`

Now that we know that, we can write:

`(2costheta)/cos^2theta`

`2/costheta=4`

`costheta/2=1/4`

`costheta=1/2`

`theta=cos^-1(1/2)`

`theta=pi/3`, when `0<=theta<=pi`.

In degrees, `theta=60^@` when `0^@<=theta<=180^@`

Answer 2

`rarrcosx=1/2`

Explanation:

Given, `rarrcosx/(1-sinx)+cosx/(1+sinx)=4`

`rarrcosx[1/(1-sinx)+1/(1+sinx)]=4`

`rarrcosx[(1+cancel(sinx)+1cancel(-sinx))/((1-sinx)*(1+sinx)]]=4`

`rarr(2cosx)/(1-sin^2x)=4`

`rarrcosx/cos^2x=2`

`rarrcosx=1/2`