Is tan(pi/2-θ)tan(θ)=1 ?

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

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prove tan(pi/2-θ)tan(θ)=1

prove

tan (\frac{π}{2} - θ) tan (θ) = 1

True

Solution steps

tan (\frac{π}{2} - θ) tan (θ) = 1

Manipulating left side

tan (\frac{π}{2} - θ) tan (θ)

Rewrite using trig identities

= \frac{cos ( θ )}{sin ( θ )} tan (θ)

Multiply fractions:

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

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

Express with sin, cos

= 1

We showed that the two sides could take the same form

\Rightarrow True

(1 - sin^{2} (x)) csc^{2} (x) = cot^{2} (x)

sin (t) tan (t) = \frac{1 - cos ^{2} ( t )}{cos ( t )}

\frac{sin ^{2} ( θ )}{cos ( θ )} = sec (θ) - cos (θ)

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

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