Question

Prove the cofunction identity using the Addition and Subtraction Formulas.

$\sec \left(\frac{\pi }{2}-u\right)=\csc u$

Short answer

$\sec \left(\frac{\pi }{2}-u\right)=\csc u$

Step-by-step solution

Step 1. Given information.

The given equation is,$\sec \left(\frac{\pi }{2}-u\right)=\csc u$

Step 2. Find the cofunction identity using the addition and subtraction formulasFind the equation,

$$\begin{gathered} \cot \left(\frac{\pi }{2}-u\right)=\frac{\cos \left(\frac{\pi }{2}-u\right)}{\sin \left(\frac{\pi }{2}-u\right)} \\ \cot x=\frac{\cos x}{\sin x} \end{gathered}$$

$$\begin{gathered} =\frac{\cos \frac{\pi }{2}\cos u+\sin \frac{\pi }{2}\sin u}{\sin \frac{\pi }{2}\cos u-\cos \frac{\pi }{2}\sin u}\text{(Additionformulasforsine,cosine)} \\ =\frac{0\cdot \cos u+1\cdot \sin u}{1\cdot \cos u-0\cdot \sin u}\cos \frac{\pi }{2}=0,\sin \frac{\pi }{2}=1 \\ =\frac{\sin u}{\cos u} \end{gathered}$$