Trigonometry Examples

$(\sec (u) - \tan (u)) (\sec (u) + \tan (u)) = 1$

Step 1

Start on the left side.

$(\sec (u) - \tan (u)) (\sec (u) + \tan (u))$

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$(\frac{1}{\cos (u)} - \tan (u)) (\sec (u) + \tan (u))$

Step 2.1.2

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

$(\frac{1}{\cos (u)} - \frac{\sin (u)}{\cos (u)}) (\sec (u) + \tan (u))$

Step 2.2

Simplify each term.

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Step 2.2.1

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

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

Step 2.2.2

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

$(\frac{1}{\cos (u)} - \frac{\sin (u)}{\cos (u)}) (\frac{1}{\cos (u)} + \frac{\sin (u)}{\cos (u)})$

Step 2.3

Expand $(\frac{1}{\cos (u)} - \frac{\sin (u)}{\cos (u)}) (\frac{1}{\cos (u)} + \frac{\sin (u)}{\cos (u)})$ using the FOIL Method.

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Step 2.3.1

Apply the distributive property.

$\frac{1}{\cos (u)} (\frac{1}{\cos (u)} + \frac{\sin (u)}{\cos (u)}) - \frac{\sin (u)}{\cos (u)} (\frac{1}{\cos (u)} + \frac{\sin (u)}{\cos (u)})$

Step 2.3.2

Apply the distributive property.

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

Step 2.3.3

Apply the distributive property.

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

Step 2.4

Combine the opposite terms in $\frac{1}{\cos (u)} \cdot \frac{1}{\cos (u)} + \frac{1}{\cos (u)} \cdot \frac{\sin (u)}{\cos (u)} - \frac{\sin (u)}{\cos (u)} \cdot \frac{1}{\cos (u)} - \frac{\sin (u)}{\cos (u)} \cdot \frac{\sin (u)}{\cos (u)}$ .

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Step 2.4.1

Reorder the factors in the terms $\frac{1}{\cos (u)} \cdot \frac{\sin (u)}{\cos (u)}$ and $- \frac{\sin (u)}{\cos (u)} \cdot \frac{1}{\cos (u)}$ .

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

Step 2.4.2

Subtract $\frac{1}{\cos (u)} \cdot \frac{\sin (u)}{\cos (u)}$ from $\frac{1}{\cos (u)} \cdot \frac{\sin (u)}{\cos (u)}$ .

$\frac{1}{\cos (u)} \cdot \frac{1}{\cos (u)} + 0 - \frac{\sin (u)}{\cos (u)} \cdot \frac{\sin (u)}{\cos (u)}$

Step 2.4.3

Add $\frac{1}{\cos (u)} \cdot \frac{1}{\cos (u)}$ and $0$ .

$\frac{1}{\cos (u)} \cdot \frac{1}{\cos (u)} - \frac{\sin (u)}{\cos (u)} \cdot \frac{\sin (u)}{\cos (u)}$

Step 2.5

Simplify each term.

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Step 2.5.1

Multiply $\frac{1}{\cos (u)} \cdot \frac{1}{\cos (u)}$ .

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Step 2.5.1.1

Multiply $\frac{1}{\cos (u)}$ by $\frac{1}{\cos (u)}$ .

$\frac{1}{\cos (u) \cos (u)} - \frac{\sin (u)}{\cos (u)} \cdot \frac{\sin (u)}{\cos (u)}$

Step 2.5.1.2

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

$\frac{1}{\cos^{1} (u) \cos (u)} - \frac{\sin (u)}{\cos (u)} \cdot \frac{\sin (u)}{\cos (u)}$

Step 2.5.1.3

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

$\frac{1}{\cos^{1} (u) \cos^{1} (u)} - \frac{\sin (u)}{\cos (u)} \cdot \frac{\sin (u)}{\cos (u)}$

Step 2.5.1.4

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

$\frac{1}{{\cos (u)}^{1 + 1}} - \frac{\sin (u)}{\cos (u)} \cdot \frac{\sin (u)}{\cos (u)}$

Step 2.5.1.5

Add $1$ and $1$ .

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

Step 2.5.2

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

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Step 2.5.2.1

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

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

Step 2.5.2.2

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

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

Step 2.5.2.3

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

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

Step 2.5.2.4

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

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

Step 2.5.2.5

Add $1$ and $1$ .

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

Step 2.5.2.6

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

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

Step 2.5.2.7

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

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

Step 2.5.2.8

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

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

Step 2.5.2.9

Add $1$ and $1$ .

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Apply pythagorean identity.

$\frac{\cos^{2} (u)}{\cos^{2} (u)}$

Step 2.8

Cancel the common factor of $\cos^{2} (u)$ .

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Step 2.8.1

Cancel the common factor.

$\frac{\cos^{2} (u)}{\cos^{2} (u)}$

Step 2.8.2

Rewrite the expression.

$1$

Step 3

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

$(\sec (u) - \tan (u)) (\sec (u) + \tan (u)) = 1$ is an identity