Trigonometry Examples

$\csc (90 ° - θ)$

Step 1

Replace $\csc (90 ° - θ)$ with an equivalent expression $\frac{1}{\sin (90 ° - θ)}$ using the fundamental identities.

$\frac{1}{\sin (90 ° - θ)}$

Step 2

Use a sum or difference formula on the denominator.

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Step 2.1

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

$\frac{1}{\sin (90 °) \cos (θ) - \cos (90 °) \sin (θ)}$

Step 2.2

Remove parentheses.

$\frac{1}{\sin (90 °) \cos (θ) - \cos (90 °) \sin (θ)}$

Step 2.3

Simplify each term.

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Step 2.3.1

The exact value of $\sin (90 °)$ is $1$ .

$\frac{1}{1 \cos (θ) - \cos (90 °) \sin (θ)}$

Step 2.3.2

Multiply $\cos (θ)$ by $1$ .

$\frac{1}{\cos (θ) - \cos (90 °) \sin (θ)}$

Step 2.3.3

The exact value of $\cos (90 °)$ is $0$ .

$\frac{1}{\cos (θ) + 0 \sin (θ)}$

Step 2.3.4

Multiply $- 1$ by $0$ .

$\frac{1}{\cos (θ) + 0 \sin (θ)}$

Step 2.3.5

Multiply $0$ by $\sin (θ)$ .

$\frac{1}{\cos (θ) + 0}$

Step 2.4

Add $\cos (θ)$ and $0$ .

$\frac{1}{\cos (θ)}$

Step 3

Convert from $\frac{1}{\cos (θ)}$ to $\sec (θ)$ .

$\sec (θ)$