## SugarRainbow 3 years ago HELP SIMPLIFY (sinx/1-cosx)+(1-cosx/sinx)

I believe you have attempted to do it. Where were you stuck? Though the answer isn't that pretty..

2. SugarRainbow

okay so i was thinking of multiplying cosx to the numerator and denominator of the (sinx/1-cosx) so then the denominator will be sin^2x and then multiply sinx to (1-cosx/sinx) so that both denominators will bbe sin^2x is that even a good first step to take?

3. seitys

sinx/(1-cosx)*(1+cosx)/(1+cosx) + (1-cosx/sinx) =sinx(1+cosx)/sin^2 x + (1-cosx/sinx) =(1+cosx/sinx)+(1-cosx/sinx) =(1+cosx+1-cosx)/sinx =2/sinx

4. SugarRainbow

why did you do this sinx(1+cosx)/sin^2 x or how'd that happen

5. SugarRainbow

oh wait nevermind i see what you did there

6. seitys

when you multiply (1-cosx) by its conjugate (1+cosx), you get (1-cos^2 x) which is just sin^2 x --> sin^2 x + cos^2 x = 1 so 1-cos^2 x=sin^2 x

7. SugarRainbow

okay well how did that turn into (1+cosx/sinx)?

8. seitys

there is a sinx in the numerator and a sin^2 x in the denominator. so 1 of them cancels out.

9. SugarRainbow

and you got 2/sinx because the +cosx and the -cosx cancel out right?

10. seitys

because that is the remaining denominator for both terms.

11. seitys

yes

12. seitys

you can simplify it even further because 1/sinx = cscx so it can also be written as 2cscx

13. SugarRainbow

oh okay thank you this was really helpful