Question

Question #38190

Answer 1

See below.

Explanation:

Rule number two or three for identities (depending on the math teacher): when you see fractions, add them. It's an extremely useful step that reveals more information than what you have to begin with.

Our common denominator for the addition here is `(sin)(1+cos)`. For the first term:
`sin/(1+cos)`

If we want `(sin)(1+cos)` on the bottom, then we need to multiply the bottom by `sin`; and what we do to the bottom, we do to the top. Therefore, this becomes:
`((sin)(sin))/((1+cos)(sin))`
`=((sin)(sin))/((1+cos)(sin))`

Note that I'm not in a hurry to simplify anything further. Leave everything by itself (for instance, don't combine `(1+cos)(sin)` to make `sin+(sin)(cos)` because this makes it more complicated than it needs to be).

Likewise, we multiply the second term by `(1+cos)/(1+cos)`, to get:
`((1+cos)(1+cos))/((sin)(1+cos))`

We now have:
`((sin)(sin))/((1+cos)(sin))+((1+cos)(1+cos))/((sin)(1+cos))`

We now add to get:
`((sin)(sin)+(1+cos)(1+cos))/((sin)(1+cos))`

Now we can multiply everything out, since we're obviously getting nowhere with the above expression:
`(sin^2+2cos+cos^2+1)/((sin)(1+cos))`

Recognize a Pythagorean Identity in here? I think this is the most challenging part of identities - at least, from what I saw from my classmates in precalc. It's kind of hard spotting identities in expressions like these, and it takes tons of practice. Recall that `sin^2+cos^2=1`; this is probably the most important identity in trig. It's also hiding in the numerator, right here:
`(color(red)(sin^2)+2cos+color(red)(cos^2)+1)/((sin)(1+cos))`

We can replace that with `1`, since it equals `1`:
`(color(red)(1)+2cos+1)/((sin)(1+cos))`

And add the two ones in the numerator to get `2`:
`(2+2cos)/((sin)(1+cos))`

Since both terms in the numerator have a common factor of two, we might try factoring out the two:
`(2(1+cos))/((sin)(1+cos))`

And what do you know, the `1+cos` cancels:
`(2cancel((1+cos)))/((sin)cancel((1+cos)))`
`=2/sin)=2csc->`since `1/sin)=csc`

And boom, we're done. Also, I've been using `sin` and `cos` to make the answer easier on the eyes, but we really should be using `sinx` and `cosx`. Doesn't matter to me, but it does matter to the people giving out the grades.