Question

How do you simplify `[(1 + sin theta)/cos theta] + [cos theta/(1 + sin theta)]`?

Answer 1

`2sectheta`

Explanation:

Multiply the fractions to achieve a common denominator.

`=[(1+sintheta)/costheta][(1+sintheta)/(1+sintheta)]+[costheta/(1+sintheta)][costheta/costheta]`

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

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

Recall the Pythagorean Identity `sin^2theta+cos^2theta=1`.

`=(2+2sintheta)/(costheta(1+sintheta))`

`=(2(1+sintheta))/(costheta(1+sintheta))`

`=2/costheta`

`=2sectheta`

Answer 2

`2sectheta`

Explanation:

Begin by writing the fractions as a single fraction by extracting the lowest common denominator
In this case `costheta(1 + sintheta)`

`rArr( (1 +sintheta)^2 + cos^2 theta)/(costheta(1+sintheta)`

Expanding the numerator

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

using the identity `(sin^2theta+cos^2theta=1)`

then `(1+2sintheta+1)/(costheta(1+sintheta)) =(2+2sintheta)/(costheta(1+sintheta))`

`=(2(1+sintheta))/(costheta(1+sintheta))=(2cancel((1+sintheta)))/(costhetacancel((1+sintheta)))`

`= 2/costheta=2sectheta`