Precalculus Examples

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

Step 1

Start on the left side.

$(1 + \sec (θ)) (1 - \sec (θ))$

Step 2

Apply the distributive property.

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

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $1 + \sec (θ)$ by $1$ .

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

Step 3.1.2

Apply the distributive property.

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

Step 3.1.3

Multiply $- \sec (θ)$ by $1$ .

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

Step 3.1.4

Multiply $\sec (θ) (- \sec (θ))$ .

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Step 3.1.4.1

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

$1 + \sec (θ) - \sec (θ) - ({\sec (θ)}^{1} \sec (θ))$

Step 3.1.4.2

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

$1 + \sec (θ) - \sec (θ) - ({\sec (θ)}^{1} {\sec (θ)}^{1})$

Step 3.1.4.3

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

$1 + \sec (θ) - \sec (θ) - {\sec (θ)}^{1 + 1}$

Step 3.1.4.4

Add $1$ and $1$ .

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

Step 3.2

Subtract $\sec (θ)$ from $\sec (θ)$ .

$1 + 0 - {\sec (θ)}^{2}$

Step 3.3

Add $1$ and $0$ .

$1 - \sec^{2} (θ)$

Step 4

Apply Pythagorean identity.

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Step 4.1

Reorder $1$ and $- \sec^{2} (θ)$ .

$- \sec^{2} (θ) + 1$

Step 4.2

Factor $- 1$ out of $- \sec^{2} (θ)$ .

$- (\sec^{2} (θ)) + 1$

Step 4.3

Rewrite $1$ as $- 1 (- 1)$ .

$- \sec^{2} (θ) - 1 \cdot - 1$

Step 4.4

Factor $- 1$ out of $- \sec^{2} (θ) - 1 \cdot - 1$ .

$- (\sec^{2} (θ) - 1)$

Step 4.5

Apply pythagorean identity.

$- \tan^{2} (θ)$

Step 5

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

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