Trigonometry Examples

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

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.

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

Step 1.2

Simplify the expression.

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Step 1.2.1

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

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

Step 1.2.2

One to any power is one.

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

Step 1.3

Apply pythagorean identity.

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

Step 1.4

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

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

$1 = 1$

Step 2

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

$x \approx 0, 0.1, - 0.3, 0.3, - \frac{2}{5}, \frac{2}{5}, \frac{1}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x \approx 0, 0.1, - 0.3, 0.3, - \frac{2}{5}, \frac{2}{5}, \frac{1}{2}$

Decimal Form:

$x \approx 0, 0.1, - 0.3, 0.3, - 0.4, 0.4, 0.5$

Step 4