Trigonometry Examples

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

Step 1

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

$\sec (x) - \sec (x) \sin^{2} (x) - \cos (x) = 0$

Step 2

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

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Step 2.1

Multiply by $1$ .

$\sec (x) \cdot 1 - \sec (x) \sin^{2} (x) - \cos (x) = 0$

Step 2.2

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

$\sec (x) \cdot 1 + \sec (x) (- 1 \sin^{2} (x)) - \cos (x) = 0$

Step 2.3

Factor $\sec (x)$ out of $\sec (x) \cdot 1 + \sec (x) (- 1 \sin^{2} (x))$ .

$\sec (x) \cdot (1 - 1 \sin^{2} (x)) - \cos (x) = 0$

Step 2.4

Rewrite $- 1 \sin^{2} (x)$ as $- \sin^{2} (x)$ .

$\sec (x) \cdot (1 - \sin^{2} (x)) - \cos (x) = 0$

Step 2.5

Apply pythagorean identity.

$\sec (x) \cdot \cos^{2} (x) - \cos (x) = 0$

Step 2.6

Simplify each term.

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Step 2.6.1

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

$\frac{1}{\cos (x)} \cdot \cos^{2} (x) - \cos (x) = 0$

Step 2.6.2

Cancel the common factor of $\cos (x)$ .

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Step 2.6.2.1

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

$\frac{1}{\cos (x)} \cdot (\cos (x) \cos (x)) - \cos (x) = 0$

Step 2.6.2.2

Cancel the common factor.

$\frac{1}{\cos (x)} \cdot (\cos (x) \cos (x)) - \cos (x) = 0$

Step 2.6.2.3

Rewrite the expression.

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

Step 2.7

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

$0 = 0$

Step 3

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:

$(- \infty, \infty)$