Trigonometry Examples

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

Step 1

Multiply both sides by $\cos^{2} (x)$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $\cos^{2} (x)$ .

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Step 2.1.1.1

Cancel the common factor.

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

Step 2.1.1.2

Rewrite the expression.

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(\sec^{2} (x) - 2 \tan^{2} (x)) \cos^{2} (x)$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 2.2.1.1.3

One to any power is one.

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

Step 2.2.1.1.4

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

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

Step 2.2.1.1.5

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

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

Step 2.2.1.1.6

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

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

Step 2.2.1.1.7

Move the negative in front of the fraction.

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

Step 2.2.1.2

Combine the numerators over the common denominator.

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

Step 2.2.1.3

Apply the cosine double-angle identity.

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

Step 2.2.1.4

Cancel the common factor of $\cos^{2} (x)$ .

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Step 2.2.1.4.1

Cancel the common factor.

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

Step 2.2.1.4.2

Rewrite the expression.

$\cos (2 x) = \cos (2 x)$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $\cos (2 x)$ to the left side of the equation.

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Step 3.1.1

Subtract $\cos (2 x)$ from both sides of the equation.

$\cos (2 x) - \cos (2 x) = 0$

Step 3.1.2

Subtract $\cos (2 x)$ from $\cos (2 x)$ .

$0 = 0$

Step 3.2

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$