Thank you for the opportunity to help you with your question!
First recognize that cotx = cosx / sinx
cot^2x + cotx / cot^2x - 1 = (cos^2x/sin^2x + cosx/sinx) / (cos^2x/sin^2x - 1)
Multiply numerator and denominator by sin^2x to eliminate sin^2x in the fractions, to get
(cos^2x + sinxcosx) / (cos^2x - sin^2x) notice the perfect square in the denominator
factor numerator and denominator:
cosx(cosx + sinx) / (cosx + sinx)(cosx - sinx)
= cosx / cosx - sinx
Now go back to the original equation with the 1 and the 3 and substitute the cot^2 fraction with cosx / cosx - sinx :
1 - cosx / cosx - sinx + 3 Now use common denominator cosx - sinx and substitute for 1 and 3 :
(cosx - sinx / cosx - sinx) - (cosx / cosx - sinx) + (3cosx - 3sinx / cosx - sinx) Now combine and simplify to get
3cosx - 4 sinx / cosx - sinx
Thank you so much, all I needed was the perfect square. I'll have to rememebr that for the future.
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