Trigonometry Examples

$1 + \frac{\tan (θ)}{\cot (θ)} = \sec^{2} (θ)$

Step 1

Start on the left side.

$1 + \frac{\tan (θ)}{\cot (θ)}$

Step 2

Simplify each term.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (θ)}{\sin (θ)}$ .

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

Step 2.4

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

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

Add $1$ and $1$ .

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

Step 2.6

Simplify the denominator.

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Add $1$ and $1$ .

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

Step 3

Apply Pythagorean identity in reverse.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \cos (θ)$ .

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

Step 4.2

Write $1$ as a fraction with a common denominator.

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 4.4

Simplify the numerator.

$\frac{1}{\cos^{2} (θ)}$

Step 5

Rewrite $\frac{1}{\cos^{2} (θ)}$ as $\sec^{2} (θ)$ .

$\sec^{2} (θ)$

Step 6

Because the two sides have been shown to be equivalent, the equation is an identity.

$1 + \frac{\tan (θ)}{\cot (θ)} = \sec^{2} (θ)$ is an identity