Is (cot(θ)-csc(θ))/(cot(θ)+csc(θ))=(1-2cos(θ)+cos^2(θ))/(-sin^2(θ)) ?

The answer to whether (cot(θ)-csc(θ))/(cot(θ)+csc(θ))=(1-2cos(θ)+cos^2(θ))/(-sin^2(θ)) is True

Popular >

prove (cot(θ)-csc(θ))/(cot(θ)+csc(θ))=(1-2cos(θ)+cos^2(θ))/(-sin^2(θ))

prove

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )} = \frac{1 - 2 cos ( θ ) + cos ^{2} ( θ )}{- sin ^{2} ( θ )}

True

Solution steps

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )} = \frac{1 - 2 cos ( θ ) + cos ^{2} ( θ )}{- sin ^{2} ( θ )}

Manipulating left side

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )}

Express with sin, cos

= \frac{- 1 + cos ( θ )}{1 + cos ( θ )}

Manipulating right side

\frac{1 - 2 cos ( θ ) + cos ^{2} ( θ )}{- sin ^{2} ( θ )}

Rewrite using trig identities

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

We showed that the two sides could take the same form

\Rightarrow True

\frac{csc ( x ) - cot ( x )}{1 - cos ( x )} = csc (x)

sin^{2} (\frac{θ}{2}) = \frac{csc ( θ ) - cot ( θ )}{2 csc ( θ )}

cos (2 x) = \frac{cot ^{2} ( x ) - 1}{cot ^{2} ( x ) + 1}

sin^{2} (x) - 2 cos^{2} (x) = 3 sin^{2} (x) - 2

cos (x) + cos (x) cot^{2} (x) = cot (x) csc (x)

Is (cot(θ)-csc(θ))/(cot(θ)+csc(θ))=(1-2cos(θ)+cos^2(θ))/(-sin^2(θ)) ?
The answer to whether (cot(θ)-csc(θ))/(cot(θ)+csc(θ))=(1-2cos(θ)+cos^2(θ))/(-sin^2(θ)) is True