Trigonometry Examples

$\sec (\frac{π}{2} - u)$

Step 1

Replace $\sec (\frac{π}{2} - u)$ with an equivalent expression $\frac{1}{\cos (\frac{π}{2} - u)}$ using the fundamental identities.

$\frac{1}{\cos (\frac{π}{2} - u)}$

Step 2

Use a sum or difference formula on the denominator.

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Step 2.1

Use the difference formula for cosine to simplify the expression. The formula states that $\cos (A - B) = \cos (A) \cos (B) + \sin (A) \sin (B)$ .

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

Step 2.2

Remove parentheses.

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

Step 2.3

Simplify each term.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.4

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

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

Step 3

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

$\csc (u)$