Trigonometry Examples

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

Step 1

Simplify $\cos (x) \cdot (\sec (x) - \cos (x))$ .

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Step 1.1

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

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

Step 1.2

Apply the distributive property.

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

Step 1.3

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

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Rewrite using the commutative property of multiplication.

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

Step 1.5

Multiply $- \cos (x) \cos (x)$ .

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Step 1.5.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.5.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.5.3

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

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

Step 1.5.4

Add $1$ and $1$ .

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

Step 1.6

Apply pythagorean identity.

$\sin^{2} (x) = \sin^{2} (x)$

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