Trigonometry Examples

$(1 + \sin (B)) (1 - \sin (B)) = \frac{1}{\sec^{2} (B)}$

Step 1

Start on the left side.

$(1 + \sin (B)) (1 - \sin (B))$

Step 2

Simplify the expression.

Tap for more steps...

Step 2.1

Expand $(1 + \sin (B)) (1 - \sin (B))$ using the FOIL Method.

Tap for more steps...

Step 2.1.1

Apply the distributive property.

$1 (1 - \sin (B)) + \sin (B) (1 - \sin (B))$

Step 2.1.2

Apply the distributive property.

$1 \cdot 1 + 1 (- \sin (B)) + \sin (B) (1 - \sin (B))$

Step 2.1.3

Apply the distributive property.

$1 \cdot 1 + 1 (- \sin (B)) + \sin (B) \cdot 1 + \sin (B) (- \sin (B))$

Step 2.2

Simplify and combine like terms.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Multiply $1$ by $1$ .

$1 + 1 (- \sin (B)) + \sin (B) \cdot 1 + \sin (B) (- \sin (B))$

Step 2.2.1.2

Multiply $- \sin (B)$ by $1$ .

$1 - \sin (B) + \sin (B) \cdot 1 + \sin (B) (- \sin (B))$

Step 2.2.1.3

Multiply $\sin (B)$ by $1$ .

$1 - \sin (B) + \sin (B) + \sin (B) (- \sin (B))$

Step 2.2.1.4

Rewrite using the commutative property of multiplication.

$1 - \sin (B) + \sin (B) - \sin (B) \sin (B)$

Step 2.2.1.5

Multiply $- \sin (B) \sin (B)$ .

Tap for more steps...

Step 2.2.1.5.1

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

$1 - \sin (B) + \sin (B) - (\sin^{1} (B) \sin (B))$

Step 2.2.1.5.2

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

$1 - \sin (B) + \sin (B) - (\sin^{1} (B) \sin^{1} (B))$

Step 2.2.1.5.3

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

$1 - \sin (B) + \sin (B) - {\sin (B)}^{1 + 1}$

Step 2.2.1.5.4

Add $1$ and $1$ .

$1 - \sin (B) + \sin (B) - \sin^{2} (B)$

Step 2.2.2

Add $- \sin (B)$ and $\sin (B)$ .

$1 + 0 - \sin^{2} (B)$

Step 2.2.3

Add $1$ and $0$ .

$1 - \sin^{2} (B)$

Step 2.3

Apply pythagorean identity.

$\cos^{2} (B)$

Step 3

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

$\frac{1}{\sec^{2} (B)}$

Step 4

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

$(1 + \sin (B)) (1 - \sin (B)) = \frac{1}{\sec^{2} (B)}$ is an identity