Prove the following identity `(1+secx+tanx)/(1+secx-tanx)=(1+sinx)/cosx`

1 Answer | Add Yours

justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The identity `(1+secx+tanx)/(1+secx-tanx)=(1+sinx)/cosx` has to be proved.

`(1+secx+tanx)/(1+secx-tanx)`

=> `(1+1/cosx+sin x/cos x)/(1+1/cos x-sin x/cos x)`

=> `(cos x+1+sin x)/(cos x+1-sin x)`

=> `((cos x+1+sin x)(cos x + 1 - sin x))/(cos x+1-sin x)^2`

=> `((cos x+1)^2 - sin^2x)/(cos x+1-sin x)^2`

=> `(cos^2 x + 2*cos x+1 - sin^2x)/(cos^2x+1+ sin^2x + 2*cosx - 2*sin x - 2*cos x*sin x)`

=> `(cos^2 x + 2*cos x+cos^2x + sin^2x - sin^2x)/(cos^2x+1+ sin^2x + 2*cosx - 2*sin x - 2*cos x*sin x)`

=> `(2*cos^2 x + 2*cos x)/(2 + 2*cosx - 2*sin x - 2*cos x*sin x)`

=> `(cos^2 x + cos x)/(1 + cosx - sin x - cos x*sin x)`

=> `(cos x(1 + cos x))/(1 + cosx - sin x(1 + cos x))`

=> `(cos x)/(1 - sin x)`

=> `(cos x(1 + sin x))/(1 - sin^2x)`

=> `(1 + sin x)/(cos x)`

This proves that `(1+secx+tanx)/(1+secx-tanx)` = `(1 + sin x)/(cos x)`

We’ve answered 318,916 questions. We can answer yours, too.

Ask a question