Trigonometry Examples

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

Step 1

Rewrite

$\csc (x)$ in terms of sines and cosines.

$\frac{1}{\sin (x)} (1 - \frac{\cos (x)}{\sec (x)})$

Step 2

Simplify each term.

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Step 2.1

Rewrite

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

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

Step 2.2

Multiply by the reciprocal of the fraction to divide by

$\frac{1}{\cos (x)}$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Raise

$\cos (x)$ to the power of

$1$ .

$\frac{1}{\sin (x)} (1 - (\cos^{1} (x) \cos (x)))$

Step 2.3.2

Raise

$\cos (x)$ to the power of

$1$ .

$\frac{1}{\sin (x)} (1 - (\cos^{1} (x) \cos^{1} (x)))$

Step 2.3.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{\sin (x)} (1 - {\cos (x)}^{1 + 1})$

Step 2.3.4

Add

$1$ and

$1$ .

$\frac{1}{\sin (x)} (1 - \cos^{2} (x))$

Step 3

Apply pythagorean identity.

$\frac{1}{\sin (x)} \sin^{2} (x)$

Step 4

Cancel the common factor of

$\sin (x)$ .

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Step 4.1

Factor

$\sin (x)$ out of

$\sin^{2} (x)$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

$\sin (x)$