Trigonometry Examples

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

Step 1

Simplify $\sec^{2} (x) (1 + \sin (2 x))$ .

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Step 1.1

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

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

Step 1.2

Simplify terms.

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Step 1.2.1

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

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

Step 1.2.2

One to any power is one.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.2.4

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

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

Step 1.2.5

Combine $\frac{1}{\cos^{2} (x)}$ and $\sin (2 x)$ .

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

Step 1.3

Simplify each term.

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Step 1.3.1

Apply the sine double-angle identity.

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

Step 1.3.2

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

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Step 1.3.2.1

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

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

Step 1.3.2.2

Cancel the common factors.

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Step 1.3.2.2.1

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

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

Step 1.3.2.2.2

Cancel the common factor.

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

Step 1.3.2.2.3

Rewrite the expression.

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

Step 1.4

Simplify each term.

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Step 1.4.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.4.2

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

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

Step 1.4.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 1.4.4

Separate fractions.

$\sec^{2} (x) + \frac{2}{1} \cdot \frac{\sin (x)}{\cos (x)} = {(1 \tan (x))}^{2}$

Step 1.4.5

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

Step 1.4.6

Divide $2$ by $1$ .

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

Step 2

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

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

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.4636476, 2.67794504, - 3.60524026, 5.81953769, - 6.74683291, 8.96113035, - 9.88842556$

Step 4