Simplify [(1+cosx)/(sinx)]+[(sinx)/(1+cosx)]

Question

I can't figure out how to simplify this correctly for my homework. Please help if you can. Thank you.

Answer 1

It might be easier to put it all together if the two terms have a common denominator. The first term, I multiplied numerator and denominator by (1 + cos x). The second term, I multiplied both by sin x.

 

As a result I get

 

[(1 + cos x)² + sin²x]/[sin x (1 + cos x)]

= 2(1 + cos x)/[sin x (1 + cos x)]

= 2 / sin x

 

That looks a lot more simple to me. :-)

 

Sanity check: I ran this through Wolfram Alpha, and got 2 csc x. Same thing. Cheers!

Answer 2

To add the two fractions, you must first find the least common denominator, which is (six)(1+cosx).  Multiply six by (1+cosx) and multiply sinx by six, then add the numerator products and divide by the least common denominator, then simplify:

 

(1 + 2cosx + cos²x + sin²x) / ((sinx)(1+cosx)) = (1 + 2cosx + 1) / ((sinx)(1+cosx)) = (2(1 + cosx)) / ((sinx)(1+cosx)) 

= 2 / sinx = 2cscx