Trigonometry Examples

\csc (90 - θ)

$\csc (90 - θ)$

Step 1

Apply the difference of angles identity.

\frac{\sec (90) \sec (θ) \csc (90) \csc (θ)}{\sec (90) \csc (θ) - \csc (90) \sec (θ)}

$\frac{\sec (90) \sec (θ) \csc (90) \csc (θ)}{\sec (90) \csc (θ) - \csc (90) \sec (θ)}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite

\sec (90)

$\sec (90)$ in terms of sines and cosines.

\frac{\frac{1}{\cos (90)} \sec (θ) \csc (90) \csc (θ)}{\sec (90) \csc (θ) - \csc (90) \sec (θ)}

$\frac{\frac{1}{\cos (90)} \sec (θ) \csc (90) \csc (θ)}{\sec (90) \csc (θ) - \csc (90) \sec (θ)}$

Step 2.2

The exact value of

\cos (90)

$\cos (90)$ is

0

$0$ .

\frac{\frac{1}{0} \sec (θ) \csc (90) \csc (θ)}{\sec (90) \csc (θ) - \csc (90) \sec (θ)}

$\frac{\frac{1}{0} \sec (θ) \csc (90) \csc (θ)}{\sec (90) \csc (θ) - \csc (90) \sec (θ)}$

Step 2.3

The expression contains a division by

0

$0$ . The expression is undefined.

Undefined

\frac{\frac{1}{0} \sec (θ) \csc (90) \csc (θ)}{\sec (90) \csc (θ) - \csc (90) \sec (θ)}

$\frac{\frac{1}{0} \sec (θ) \csc (90) \csc (θ)}{\sec (90) \csc (θ) - \csc (90) \sec (θ)}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite

\sec (90)

$\sec (90)$ in terms of sines and cosines.

\frac{\frac{1}{0} \sec (θ) \csc (90) \csc (θ)}{\frac{1}{\cos (90)} \csc (θ) - \csc (90) \sec (θ)}

$\frac{\frac{1}{0} \sec (θ) \csc (90) \csc (θ)}{\frac{1}{\cos (90)} \csc (θ) - \csc (90) \sec (θ)}$

Step 3.2

The exact value of

\cos (90)

$\cos (90)$ is

0

$0$ .

\frac{\frac{1}{0} \sec (θ) \csc (90) \csc (θ)}{\frac{1}{0} \csc (θ) - \csc (90) \sec (θ)}

$\frac{\frac{1}{0} \sec (θ) \csc (90) \csc (θ)}{\frac{1}{0} \csc (θ) - \csc (90) \sec (θ)}$

Step 3.3

The expression contains a division by

0

$0$ . The expression is undefined.

Undefined

\frac{\frac{1}{0} \sec (θ) \csc (90) \csc (θ)}{\frac{1}{0} \csc (θ) - \csc (90) \sec (θ)}

$\frac{\frac{1}{0} \sec (θ) \csc (90) \csc (θ)}{\frac{1}{0} \csc (θ) - \csc (90) \sec (θ)}$

Step 4

The expression contains a division by

0

$0$ . The expression is undefined.

Undefined