Question

How can I simplify this trigonometry expression? Thanks!

Answer 1

See below

Explanation:

Whenever I see two fractions being added or subtracted with different denominators, I automatically think we need a common denominator.

`1/(1+sintheta)+1/(1-sintheta)=`
`(1(1-sintheta))/((1+sintheta)(1-sintheta))+(1(1+sintheta))/((1-sintheta)(1+sintheta))=`
`2/(1-sin^2theta)`
Now that we successfully added the two fractions, we will apply a Pythagorean identity to simplify:
`sin^2theta+cos^2theta=1`
Therefore:
`cos^2theta=1-sin^2theta`
Let's substitute `cos^2theta` in our denominator:
`2/(cos^2theta)`
Now apply the reciprocal identity:
`Sectheta=1/costheta`
Final answer: `2sec^2theta`

Answer 2

`2 sec^2 theta`

Explanation:

`1/(1 + sin theta) + 1/(1 - sin theta)`

multiplying each term by one in the form of a ratio of an appropriately chosen numerator and denominator:

`= (1 - sin theta)/((1 + sin theta)(1 - sin theta)) + (1 + sin theta)/((1 - sin theta)(1 + sin theta))`

using the corollary of the difference of two squares for the terms in the denominators:

`= (1 - sin theta)/(1 - sin^2theta) + (1 + sin theta)/(1 - sin^2theta)`

summing terms with the same denominator

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

using a rearrangement of the identity `cos^2 theta + sin^2 theta = 1`:

`= 2 / cos^2theta`

noting `sec^2 theta = 1/(cos^2 theta)`:

`= 2 sec^2 theta`