How do you verify the identity #(cot^2x-1)/(1+cot^2x)=1-2sin^2x#?
1 Answer
see explanation.
Explanation:
We attempt to express the left side in the same form as the right side.
Let's begin by rewriting#cot^2x#
#color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(cotx=(cosx)/(sinx))color(white)(a/a)|)))#
#rArr((cos^2x)/(sin^2x)-1)/(1+cos^2x/(sin^2x)# now multiply all terms on numerator and denominator by
#sin^2x#
#rArr(cos^2x-sin^2x)/(sin^2x+cos^2x) .......(A)# To simplify we require the
#color(blue)"trigonometric identities"#
#color(orange)"Reminder"#
#color(red)(|bar(ul(color(white)(a/a)color(black)(cos2x=cos^2x-sin^2x=1-2sin^2x)color(white)(a/a)|)))# and
#color(red)(|bar(ul(color(white)(a/a)color(black)(sin^2x+cos^2x=1)color(white)(a/a)|)))# Substituting these into (A) gives.
#cos2x=1-2sin^2x# Thus left side = right side
#rArr" verified"#
