Prove that (cosxcotx)/(1 - sinx) - 1 = cscx?
2 Answers
We start with:
color(green)((cosxcotx)/(1-sinx) - 1 stackrel(?)(=) cscx).
First, recall
(cosx*cosx/sinx)/(1-sinx) - 1 = cscx
(cos^2x/sinx)/(1-sinx) - 1 = cscx
On the fraction you can move the middle
cos^2x/(sinx(1-sinx)) - 1 = cscx
Now, when you have a
cos^2x/(sinx(1-sinx)) - (sinx(1-sinx))/(sinx(1-sinx)) = cscx
Distribute the numerator and combine into one fraction:
(-sinx + cos^2x + sin^2x)/(sinx(1-sinx)) = cscx
Then recall
cancel(1 - sinx)/(sinxcancel((1-sinx))) = cscx
1/sinx = cscx
color(blue)(cscx = cscx)
It is a bit long but...
Explanation:
You can try changing all in
Your identity becomes:
we use
and:
but:
so we get: