Trigonometry Examples

{cos}^{2} (x) (1 + {tan}^{2} (x)) = 1

$\cos^{2} (x) (1 + \tan^{2} (x)) = 1$

Step 1

Start on the left side.

{cos}^{2} (x) (1 + {tan}^{2} (x))

$\cos^{2} (x) (1 + \tan^{2} (x))$

Step 2

Apply Pythagorean identity.

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Step 2.1

Rearrange terms.

{cos}^{2} (x) ({tan}^{2} (x) + 1)

$\cos^{2} (x) (\tan^{2} (x) + 1)$

Step 2.2

Apply pythagorean identity.

{cos}^{2} (x) {sec}^{2} (x)

$\cos^{2} (x) \sec^{2} (x)$

{cos}^{2} (x) {sec}^{2} (x)

$\cos^{2} (x) \sec^{2} (x)$

Step 3

Convert to sines and cosines.

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Step 3.1

Apply the reciprocal identity to

sec (x)

$\sec (x)$ .

{cos}^{2} (x) {(\frac{1}{cos (x)})}^{2}

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

Step 3.2

Apply the product rule to

\frac{1}{cos (x)}

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

{cos}^{2} (x) \frac{1^{2}}{{cos}^{2} (x)}

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

{cos}^{2} (x) \frac{1^{2}}{{cos}^{2} (x)}

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

Step 4

Simplify.

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Step 4.1

One to any power is one.

{cos (x)}^{2} \frac{1}{{cos (x)}^{2}}

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

Step 4.2

Cancel the common factor of

{cos (x)}^{2}

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

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Step 4.2.1

Cancel the common factor.

{cos (x)}^{2} \frac{1}{{cos (x)}^{2}}

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

Step 4.2.2

Rewrite the expression.

1

$1$

1

$1$

1

$1$

Step 5

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

{cos}^{2} (x) (1 + {tan}^{2} (x)) = 1

$\cos^{2} (x) (1 + \tan^{2} (x)) = 1$ is an identity