Precalculus Examples

$\sec (x - \frac{π}{2}) = \csc (x)$

Step 1

Start on the left side.

$\sec (x - \frac{π}{2})$

Step 2

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

$\frac{1}{\cos (x - \frac{π}{2})}$

Step 3

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

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

Step 4

Simplify the expression.

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Step 4.1

Simplify the denominator.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

$\frac{1}{0 + \sin (x) \cdot 1}$

Step 4.1.4

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

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

Step 4.1.5

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

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

Step 4.2

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

$\csc (x)$

Step 5

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

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