Trigonometry Examples

$\sec (θ) \cot (θ) - \cot (θ) \cos (θ) = \sin (θ)$

Step 1

Simplify $\sec (θ) \cot (θ) - \cot (θ) \cos (θ)$ .

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Step 1.1

Simplify terms.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.1.1.1.1

Reorder $\sec (θ)$ and $\cot (θ)$ .

$\cot (θ) \sec (θ) - \cot (θ) \cos (θ) = \sin (θ)$

Step 1.1.1.1.2

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

$\frac{\cos (θ)}{\sin (θ)} \cdot \frac{1}{\cos (θ)} - \cot (θ) \cos (θ) = \sin (θ)$

Step 1.1.1.1.3

Cancel the common factors.

$\frac{1}{\sin (θ)} - \cot (θ) \cos (θ) = \sin (θ)$

Step 1.1.1.2

Rewrite $\cot (θ)$ in terms of sines and cosines.

$\frac{1}{\sin (θ)} - \frac{\cos (θ)}{\sin (θ)} \cos (θ) = \sin (θ)$

Step 1.1.1.3

Multiply $- \frac{\cos (θ)}{\sin (θ)} \cos (θ)$ .

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Step 1.1.1.3.1

Combine $\cos (θ)$ and $\frac{\cos (θ)}{\sin (θ)}$ .

$\frac{1}{\sin (θ)} - \frac{\cos (θ) \cos (θ)}{\sin (θ)} = \sin (θ)$

Step 1.1.1.3.2

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

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

Step 1.1.1.3.3

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

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

Step 1.1.1.3.4

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

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

Step 1.1.1.3.5

Add $1$ and $1$ .

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

Step 1.1.2

Combine the numerators over the common denominator.

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

Step 1.2

Apply pythagorean identity.

$\frac{\sin^{2} (θ)}{\sin (θ)} = \sin (θ)$

Step 1.3

Cancel the common factor of $\sin^{2} (θ)$ and $\sin (θ)$ .

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Step 1.3.1

Factor $\sin (θ)$ out of $\sin^{2} (θ)$ .

$\frac{\sin (θ) \sin (θ)}{\sin (θ)} = \sin (θ)$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Multiply by $1$ .

$\frac{\sin (θ) \sin (θ)}{\sin (θ) \cdot 1} = \sin (θ)$

Step 1.3.2.2

Cancel the common factor.

$\frac{\sin (θ) \sin (θ)}{\sin (θ) \cdot 1} = \sin (θ)$

Step 1.3.2.3

Rewrite the expression.

$\frac{\sin (θ)}{1} = \sin (θ)$

Step 1.3.2.4

Divide $\sin (θ)$ by $1$ .

$\sin (θ) = \sin (θ)$

Step 2

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

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

Step 3