Prove that `(1+sinx)/(cosx) + (cosx)/(1-sinx) = 2sec x`

Expert Answers

An illustration of the letter 'A' in a speech bubbles

`(1+sinx)/(cos x)+ (cosx)/(1-sinx) = 2 sec x`

Let's simplify left side of the equation.

Since the denominators are cos x and 1-sin x, the LCD is cosx(1-sinx).

`(1+sinx)/(cosx) *(1-sinx)/(1-sinx) + (cosx)/(1-sinx)*(cosx)/(cosx)`

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

Base on the Pythagorean identity, `cos^2x = 1- sin^2x` .

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

Add the fractions.

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

Factor  `1-sin^2x` .

`= (2(1-sinx)(1+sinx))/(cosx(1- sinx))`

Cancel common factor between numerator and denominator.

`= (2 (1+sinx))/(cosx)`

`= (2+2sinx)/(cosx)`

Express as two fractions.

`= 2/(cosx) + (2sinx)/(cosx)`

Note that `sec x = 1/(cos x)`   and   `tanx =(sinx)/(cosx)` .

`= 2secx + 2tanx`

Since the simplified form of the left side is not the same with the given expression at the right side of the equation, hence

`(1+sinx)/(cosx) + (cosx)/(1-sinx)`  `!=`  `2secx` .

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial