Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

$1 - (\cos (x) \cos (x))$

Step 2.1.3

Simplify.

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Step 2.1.3.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 2.1.3.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 2.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.1.3.4

Add $1$ and $1$ .

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

Step 2.2

Apply pythagorean identity.

$\sin^{2} (x)$

Step 3

Rewrite $\sin^{2} (x)$ as $\frac{\sin (x)}{\csc (x)}$ .

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

Step 4

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

$1 - \frac{\cos (x)}{\sec (x)} = \frac{\sin (x)}{\csc (x)}$ is an identity