Trigonometry Examples

$\frac{\cos (t)}{1 + \sin (t)} + \frac{1 + \sin (t)}{\cos (t)} = 2 \sec (t)$

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

Tap for more steps...

Step 2.1

To write $\frac{\cos (t)}{1 + \sin (t)}$ as a fraction with a common denominator, multiply by $\frac{\cos (t)}{\cos (t)}$ .

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

Step 2.2

To write $\frac{1 + \sin (t)}{\cos (t)}$ as a fraction with a common denominator, multiply by $\frac{1 + \sin (t)}{1 + \sin (t)}$ .

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

Step 2.3

Write each expression with a common denominator of $(1 + \sin (t)) \cos (t)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.3.1

Multiply $\frac{\cos (t)}{1 + \sin (t)}$ by $\frac{\cos (t)}{\cos (t)}$ .

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

Step 2.3.2

Multiply $\frac{1 + \sin (t)}{\cos (t)}$ by $\frac{1 + \sin (t)}{1 + \sin (t)}$ .

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

Step 2.3.3

Reorder the factors of $(1 + \sin (t)) \cos (t)$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

Tap for more steps...

Step 2.5.1

Multiply $\cos (t) \cos (t)$ .

Tap for more steps...

Step 2.5.1.1

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

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

Step 2.5.1.2

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

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

Step 2.5.1.3

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

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

Step 2.5.1.4

Add $1$ and $1$ .

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

Step 2.5.2

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

Tap for more steps...

Step 2.5.2.1

Apply the distributive property.

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

Step 2.5.2.2

Apply the distributive property.

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

Step 2.5.2.3

Apply the distributive property.

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

Step 2.5.3

Simplify and combine like terms.

Tap for more steps...

Step 2.5.3.1

Simplify each term.

Tap for more steps...

Step 2.5.3.1.1

Multiply $1$ by $1$ .

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

Step 2.5.3.1.2

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

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

Step 2.5.3.1.3

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

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

Step 2.5.3.1.4

Multiply $\sin (t) \sin (t)$ .

Tap for more steps...

Step 2.5.3.1.4.1

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

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

Step 2.5.3.1.4.2

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

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

Step 2.5.3.1.4.3

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

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

Step 2.5.3.1.4.4

Add $1$ and $1$ .

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

Step 2.5.3.2

Add $\sin (t)$ and $\sin (t)$ .

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

Step 2.5.4

Rewrite $\cos^{2} (t) + 1 + 2 \sin (t) + \sin^{2} (t)$ in a factored form.

Tap for more steps...

Step 2.5.4.1

Regroup terms.

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

Step 2.5.4.2

Rearrange terms.

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

Step 2.5.4.3

Apply pythagorean identity.

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

Step 2.5.4.4

Add $1$ and $1$ .

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

Step 2.5.4.5

Factor $2$ out of $2 + 2 \sin (t)$ .

Tap for more steps...

Step 2.5.4.5.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 1 + 2 \sin (t)}{\cos (t) (1 + \sin (t))}$

Step 2.5.4.5.2

Factor $2$ out of $2 \cdot 1 + 2 \sin (t)$ .

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

Step 2.6

Cancel the common factor of $1 + \sin (t)$ .

Tap for more steps...

Step 2.6.1

Cancel the common factor.

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

Step 2.6.2

Rewrite the expression.

$\frac{2}{\cos (t)}$

Step 3

Rewrite $\frac{2}{\cos (t)}$ as $2 \sec (t)$ .

$2 \sec (t)$

Step 4

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

$\frac{\cos (t)}{1 + \sin (t)} + \frac{1 + \sin (t)}{\cos (t)} = 2 \sec (t)$ is an identity