Question

Prove the cofunction identity using the Addition and Subtraction Formulas.

$\tan \left(\frac{\pi }{2}-u\right)=\cot u$

Short answer

$\tan \left(\frac{\pi }{2}-u\right)=\cot u$

Step-by-step solution

Step 1.The given equation.

The given equation is$\tan \left(\frac{\pi }{2}-u\right)=\cot u$

Step 2. Find the cofunction identity using the addition and subtraction formulas.

Simplify the equation,

Use the Subtraction formula for Sines and Cosines to expand the numerator and denominator.

Note that $\sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \alpha$

$\cos (\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta$

Where role="math" localid="1648731327285" $\alpha =\frac{\pi }{2}$and $\beta =u.$
Recall that, $\sin \frac{\pi }{2}=1,\cos \frac{\pi }{2}=0$and $\cot \theta =\frac{\cos \theta }{\sin \theta }$.

Find the cofunction identity,

$\begin{array}{l}\tan \left(\frac{\pi }{2}-u\right)\equiv \frac{\sin \frac{\pi }{2}\cos u-\cos \frac{\pi }{2}\sin u}{\cos \frac{\pi }{2}\cos u+\sin \frac{\pi }{2}\sin u}\end{array}$

$\begin{array}{l}\equiv \frac{\left(1\right)\left(\cos u\right)-\left(0\right)\left(\sin u\right)}{\left(0\right)\left(\cos u\right)+\left(1\right)\left(\sin u\right)}\\ =\frac{\cos u}{\sin u}\\ =\cot u\end{array}$