Trigonometry Examples

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

Step 1

Start on the right side.

${(\frac{\sec (t) + \csc (t)}{\sec (t) \csc (t)})}^{2}$

Step 2

Convert to sines and cosines.

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Step 2.1

Apply the reciprocal identity to $\sec (t)$ .

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

Step 2.2

Apply the reciprocal identity to $\csc (t)$ .

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

Step 2.3

Apply the reciprocal identity to $\sec (t)$ .

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

Step 2.4

Apply the reciprocal identity to $\csc (t)$ .

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

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

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

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

Step 2.5.1.2

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

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

Step 2.5.1.3

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

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Step 2.5.1.3.1

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

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

Step 2.5.1.3.2

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

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

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

Step 2.5.1.4

Combine the numerators over the common denominator.

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

Step 2.5.2

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

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

Step 2.5.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5.4

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

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Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

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

Step 2.5.5

Rewrite ${(\sin (t) + \cos (t))}^{2}$ as $(\sin (t) + \cos (t)) (\sin (t) + \cos (t))$ .

$(\sin (t) + \cos (t)) (\sin (t) + \cos (t))$

Step 2.5.6

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

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Step 2.5.6.1

Apply the distributive property.

$\sin (t) (\sin (t) + \cos (t)) + \cos (t) (\sin (t) + \cos (t))$

Step 2.5.6.2

Apply the distributive property.

$\sin (t) \sin (t) + \sin (t) \cos (t) + \cos (t) (\sin (t) + \cos (t))$

Step 2.5.6.3

Apply the distributive property.

$\sin (t) \sin (t) + \sin (t) \cos (t) + \cos (t) \sin (t) + \cos (t) \cos (t)$

Step 2.5.7

Simplify and combine like terms.

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

Step 3

Apply Pythagorean identity.

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Step 3.1

Move $\cos^{2} (t)$ .

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

Step 3.2

Apply pythagorean identity.

$1 + 2 \cos (t) \sin (t)$

Step 4

Rewrite $1 + 2 \cos (t) \sin (t)$ as $2 \sin (t) \cos (t) + 1$ .

$2 \sin (t) \cos (t) + 1$

Step 5

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

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