Trigonometry Examples

$\tan^{4} (x) + 2 \tan^{2} (x) + 1 = \sec^{4} (x)$

Step 1

Start on the left side.

$\tan^{4} (x) + 2 \tan^{2} (x) + 1$

Step 2

Convert to sines and cosines.

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Step 2.1

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

${(\frac{\sin (x)}{\cos (x)})}^{4} + 2 \tan^{2} (x) + 1$

Step 2.2

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

${(\frac{\sin (x)}{\cos (x)})}^{4} + 2 {(\frac{\sin (x)}{\cos (x)})}^{2} + 1$

Step 2.3

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\sin^{4} (x)}{\cos^{4} (x)} + 2 {(\frac{\sin (x)}{\cos (x)})}^{2} + 1$

Step 2.4

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\sin^{4} (x)}{\cos^{4} (x)} + 2 \frac{\sin^{2} (x)}{\cos^{2} (x)} + 1$

Step 3

Simplify.

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Step 3.1

Combine $2$ and $\frac{{\sin (x)}^{2}}{{\cos (x)}^{2}}$ .

$\frac{{\sin (x)}^{4}}{{\cos (x)}^{4}} + \frac{2 {\sin (x)}^{2}}{{\cos (x)}^{2}} + 1$

Step 3.2

To write $\frac{2 {\sin (x)}^{2}}{{\cos (x)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{{\cos (x)}^{2}}{{\cos (x)}^{2}}$ .

$\frac{{\sin (x)}^{4}}{{\cos (x)}^{4}} + \frac{2 {\sin (x)}^{2}}{{\cos (x)}^{2}} \cdot \frac{{\cos (x)}^{2}}{{\cos (x)}^{2}} + 1$

Step 3.3

Write each expression with a common denominator of ${\cos (x)}^{4}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{2 {\sin (x)}^{2}}{{\cos (x)}^{2}}$ by $\frac{{\cos (x)}^{2}}{{\cos (x)}^{2}}$ .

$\frac{{\sin (x)}^{4}}{{\cos (x)}^{4}} + \frac{2 {\sin (x)}^{2} {\cos (x)}^{2}}{{\cos (x)}^{2} {\cos (x)}^{2}} + 1$

Step 3.3.2

Multiply ${\cos (x)}^{2}$ by ${\cos (x)}^{2}$ by adding the exponents.

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Step 3.3.2.1

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

$\frac{{\sin (x)}^{4}}{{\cos (x)}^{4}} + \frac{2 {\sin (x)}^{2} {\cos (x)}^{2}}{{\cos (x)}^{2 + 2}} + 1$

Step 3.3.2.2

Add $2$ and $2$ .

$\frac{{\sin (x)}^{4}}{{\cos (x)}^{4}} + \frac{2 {\sin (x)}^{2} {\cos (x)}^{2}}{{\cos (x)}^{4}} + 1$

Step 3.4

Combine the numerators over the common denominator.

$\frac{{\sin (x)}^{4} + 2 {\sin (x)}^{2} {\cos (x)}^{2}}{{\cos (x)}^{4}} + 1$

Step 3.5

Factor ${\sin (x)}^{2}$ out of ${\sin (x)}^{4} + 2 {\sin (x)}^{2} {\cos (x)}^{2}$ .

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Step 3.5.1

Factor ${\sin (x)}^{2}$ out of ${\sin (x)}^{4}$ .

$\frac{{\sin (x)}^{2} {\sin (x)}^{2} + 2 {\sin (x)}^{2} {\cos (x)}^{2}}{{\cos (x)}^{4}} + 1$

Step 3.5.2

Factor ${\sin (x)}^{2}$ out of $2 {\sin (x)}^{2} {\cos (x)}^{2}$ .

$\frac{{\sin (x)}^{2} {\sin (x)}^{2} + {\sin (x)}^{2} (2 {\cos (x)}^{2})}{{\cos (x)}^{4}} + 1$

Step 3.5.3

Factor ${\sin (x)}^{2}$ out of ${\sin (x)}^{2} {\sin (x)}^{2} + {\sin (x)}^{2} (2 {\cos (x)}^{2})$ .

$\frac{{\sin (x)}^{2} ({\sin (x)}^{2} + 2 {\cos (x)}^{2})}{{\cos (x)}^{4}} + 1$

Step 3.6

Write $1$ as a fraction with a common denominator.

$\frac{{\sin (x)}^{2} ({\sin (x)}^{2} + 2 {\cos (x)}^{2})}{{\cos (x)}^{4}} + \frac{{\cos (x)}^{4}}{{\cos (x)}^{4}}$

Step 3.7

Combine the numerators over the common denominator.

$\frac{{\sin (x)}^{2} ({\sin (x)}^{2} + 2 {\cos (x)}^{2}) + {\cos (x)}^{4}}{{\cos (x)}^{4}}$

Step 3.8

Simplify the numerator.

$\frac{{(\sin^{2} (x) + \cos^{2} (x))}^{2}}{\cos^{4} (x)}$

Step 4

Apply pythagorean identity.

$\frac{1^{2}}{\cos^{4} (x)}$

Step 5

One to any power is one.

$\frac{1}{\cos^{4} (x)}$

Step 6

Rewrite $\frac{1}{\cos^{4} (x)}$ as $\sec^{4} (x)$ .

$\sec^{4} (x)$

Step 7

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

$\tan^{4} (x) + 2 \tan^{2} (x) + 1 = \sec^{4} (x)$ is an identity