Expert: Ahmed Salami - 10/14/2009

Question
I have a problem here and its asking to write the following trigonometric expression in terms of sine and cosine and then simplify: sin^2x(1+cot^2x) so far i have gone back and forth but i came with no answer yet. Thanks in advance.

Answer
Hi Lucy,
Lets start with the identity cos²x + sin²x = 1 which i believe you should no doubt know if you're into trigonometry. And remember that
tanx = sinx/cosx
cosecx = 1/sinx
secx = 1/cosx
cotx = 1/tanx = cosx/sinx
Now, dividing through cos²x + sin²x = 1 by sin²x, we have
cos²x/sin²x + sin²x/sin²x = 1/sin²x
(cosx/sinx)² + 1 = 1/sin²x
cot²x + 1 = 1/sin²x
cot²x + 1 = 1/sin²x
Back to the problem,
(sin²x)(1 + cot²x) = (sin²x)(1/sin²x)
                  = 1

Of course its easy and could even be easier.

Regards