Trigonometry Examples

\sin (θ) + \cos (θ) \cot (θ)

$\sin (θ) + \cos (θ) \cot (θ)$

Step 1

Simplify each term.

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Step 1.1

Rewrite

\cot (θ)

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

\sin (θ) + \cos (θ) \frac{\cos (θ)}{\sin (θ)}

$\sin (θ) + \cos (θ) \frac{\cos (θ)}{\sin (θ)}$

Step 1.2

Multiply

\cos (θ) \frac{\cos (θ)}{\sin (θ)}

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

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Step 1.2.1

Combine

\cos (θ)

$\cos (θ)$ and

\frac{\cos (θ)}{\sin (θ)}

$\frac{\cos (θ)}{\sin (θ)}$ .

\sin (θ) + \frac{\cos (θ) \cos (θ)}{\sin (θ)}

$\sin (θ) + \frac{\cos (θ) \cos (θ)}{\sin (θ)}$

Step 1.2.2

Raise

\cos (θ)

$\cos (θ)$ to the power of

1

$1$ .

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

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

Step 1.2.3

Raise

\cos (θ)

$\cos (θ)$ to the power of

1

$1$ .

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

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

Step 1.2.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 1.2.5

Add

1

$1$ and

1

$1$ .

\sin (θ) + \frac{\cos^{2} (θ)}{\sin (θ)}

$\sin (θ) + \frac{\cos^{2} (θ)}{\sin (θ)}$

\sin (θ) + \frac{\cos^{2} (θ)}{\sin (θ)}

$\sin (θ) + \frac{\cos^{2} (θ)}{\sin (θ)}$

\sin (θ) + \frac{\cos^{2} (θ)}{\sin (θ)}

$\sin (θ) + \frac{\cos^{2} (θ)}{\sin (θ)}$

Step 2

Simplify each term.

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Step 2.1

Factor

\cos (θ)

$\cos (θ)$ out of

\cos^{2} (θ)

$\cos^{2} (θ)$ .

\sin (θ) + \frac{\cos (θ) \cos (θ)}{\sin (θ)}

$\sin (θ) + \frac{\cos (θ) \cos (θ)}{\sin (θ)}$

Step 2.2

Separate fractions.

\sin (θ) + \frac{\cos (θ)}{1} \cdot \frac{\cos (θ)}{\sin (θ)}

$\sin (θ) + \frac{\cos (θ)}{1} \cdot \frac{\cos (θ)}{\sin (θ)}$

Step 2.3

Convert from

\frac{\cos (θ)}{\sin (θ)}

$\frac{\cos (θ)}{\sin (θ)}$ to

\cot (θ)

$\cot (θ)$ .

\sin (θ) + \frac{\cos (θ)}{1} \cot (θ)

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

Step 2.4

Divide

\cos (θ)

$\cos (θ)$ by

1

$1$ .

\sin (θ) + \cos (θ) \cot (θ)

$\sin (θ) + \cos (θ) \cot (θ)$

\sin (θ) + \cos (θ) \cot (θ)

$\sin (θ) + \cos (θ) \cot (θ)$