Trigonometry Examples

$(\sec (x) - 1) (\sec (x) + 1) = \tan^{2} (x)$

Step 1

Start on the left side.

$(\sec (x) - 1) (\sec (x) + 1)$

Step 2

Apply the distributive property.

$(\sec (x) - 1) \sec (x) + (\sec (x) - 1) \cdot 1$

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the distributive property.

$\sec (x) \sec (x) - 1 \sec (x) + (\sec (x) - 1) \cdot 1$

Step 3.1.2

Multiply $\sec (x) \sec (x)$ .

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Step 3.1.2.1

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

${\sec (x)}^{1} \sec (x) - 1 \sec (x) + (\sec (x) - 1) \cdot 1$

Step 3.1.2.2

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

${\sec (x)}^{1} {\sec (x)}^{1} - 1 \sec (x) + (\sec (x) - 1) \cdot 1$

Step 3.1.2.3

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

${\sec (x)}^{1 + 1} - 1 \sec (x) + (\sec (x) - 1) \cdot 1$

Step 3.1.2.4

Add $1$ and $1$ .

${\sec (x)}^{2} - 1 \sec (x) + (\sec (x) - 1) \cdot 1$

Step 3.1.3

Rewrite $- 1 \sec (x)$ as $- \sec (x)$ .

${\sec (x)}^{2} - \sec (x) + (\sec (x) - 1) \cdot 1$

Step 3.1.4

Multiply $\sec (x) - 1$ by $1$ .

${\sec (x)}^{2} - \sec (x) + \sec (x) - 1$

Step 3.2

Add $- \sec (x)$ and $\sec (x)$ .

${\sec (x)}^{2} + 0 - 1$

Step 3.3

Add ${\sec (x)}^{2}$ and $0$ .

$\sec^{2} (x) - 1$

Step 4

Apply pythagorean identity.

$\tan^{2} (x)$

Step 5

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

$(\sec (x) - 1) (\sec (x) + 1) = \tan^{2} (x)$ is an identity