Trigonometry Examples

$\frac{\sec^{2} (x) - \tan^{2} (x) + \tan (x)}{\sec (x)} = \sin (x) + \cos (x)$

Step 1

Start on the left side.

$\frac{\sec^{2} (x) - \tan^{2} (x) + \tan (x)}{\sec (x)}$

Step 2

Apply pythagorean identity.

$\frac{1 + \tan (x)}{\sec (x)}$

Step 3

Convert to sines and cosines.

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Step 3.1

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 3.2

Apply the reciprocal identity to $\sec (x)$ .

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

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Multiply $\cos (x)$ by $1$ .

$\cos (x) + \frac{\sin (x)}{\cos (x)} \cos (x)$

Step 4.4

Cancel the common factor of $\cos (x)$ .

$\cos (x) + \sin (x)$

Step 5

Rewrite $\cos (x) + \sin (x)$ as $\sin (x) + \cos (x)$ .

$\sin (x) + \cos (x)$

Step 6

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

$\frac{\sec^{2} (x) - \tan^{2} (x) + \tan (x)}{\sec (x)} = \sin (x) + \cos (x)$ is an identity