Trigonometry Examples

$\sec (\frac{π}{2} - u) = \csc (u)$

Step 1

Start on the left side.

$\sec (\frac{π}{2} - u)$

Step 2

Apply the reciprocal identity to $\sec (\frac{π}{2} - u)$ .

$\frac{1}{\cos (\frac{π}{2} - u)}$

Step 3

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$\frac{1}{\cos (\frac{π}{2}) \cos (u) + \sin (\frac{π}{2}) \sin (u)}$

Step 4

Simplify the expression.

Tap for more steps...

Step 4.1

Simplify the denominator.

Tap for more steps...

Step 4.1.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{1}{0 \cos (u) + \sin (\frac{π}{2}) \sin (u)}$

Step 4.1.2

Multiply $0$ by $\cos (u)$ .

$\frac{1}{0 + \sin (\frac{π}{2}) \sin (u)}$

Step 4.1.3

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{1}{0 + 1 \sin (u)}$

Step 4.1.4

Multiply $\sin (u)$ by $1$ .

$\frac{1}{0 + \sin (u)}$

Step 4.1.5

Add $0$ and $\sin (u)$ .

$\frac{1}{\sin (u)}$

Step 4.2

Convert from $\frac{1}{\sin (u)}$ to $\csc (u)$ .

$\csc (u)$

Step 5

Because the two sides have been shown to be equivalent, the equation is an identity.

$\sec (\frac{π}{2} - u) = \csc (u)$ is an identity