Is cot(pi/2-θ)cot(θ)=1 ?

The answer to whether cot(pi/2-θ)cot(θ)=1 is True

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prove cot(pi/2-θ)cot(θ)=1

prove

cot (\frac{π}{2} - θ) cot (θ) = 1

True

Solution steps

cot (\frac{π}{2} - θ) cot (θ) = 1

Manipulating left side

cot (\frac{π}{2} - θ) cot (θ)

Rewrite using trig identities

= \frac{sin ( θ )}{cos ( θ )} cot (θ)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ( θ ) cot ( θ )}{cos ( θ )}

Express with sin, cos

= 1

We showed that the two sides could take the same form

\Rightarrow True

cos (\frac{x}{2}) (1 - cos (x)) = sin (\frac{x}{2}) sin (x)

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

\frac{csc ^{2} ( t )}{cot ( t )} = tan (t) + cot (t)

\frac{2}{sin ( θ )} = 2 csc (θ)

cos (x) (1 - sin^{2} (x)) = cos^{3} (x)