Trigonometry Examples

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

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

Step 1

Start on the right side.

{tan}^{2} (x) + 2 tan (x) sec (x) + {sec}^{2} (x)

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

Step 2

Simplify each term.

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Step 2.1

Rewrite

tan (x)

$\tan (x)$ in terms of sines and cosines.

{(\frac{sin (x)}{cos (x)})}^{2} + 2 tan (x) sec (x) + {sec}^{2} (x)

${(\frac{\sin (x)}{\cos (x)})}^{2} + 2 \tan (x) \sec (x) + \sec^{2} (x)$

Step 2.2

Apply the product rule to

\frac{sin (x)}{cos (x)}

$\frac{\sin (x)}{\cos (x)}$ .

\frac{{sin}^{2} (x)}{{cos}^{2} (x)} + 2 tan (x) sec (x) + {sec}^{2} (x)

$\frac{\sin^{2} (x)}{\cos^{2} (x)} + 2 \tan (x) \sec (x) + \sec^{2} (x)$

Step 2.3

Rewrite

tan (x)

$\tan (x)$ in terms of sines and cosines.

\frac{{sin}^{2} (x)}{{cos}^{2} (x)} + 2 \frac{sin (x)}{cos (x)} sec (x) + {sec}^{2} (x)

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

Step 2.4

Combine

2

$2$ and

\frac{sin (x)}{cos (x)}

$\frac{\sin (x)}{\cos (x)}$ .

\frac{{sin}^{2} (x)}{{cos}^{2} (x)} + \frac{2 sin (x)}{cos (x)} sec (x) + {sec}^{2} (x)

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

Step 2.5

Rewrite

sec (x)

$\sec (x)$ in terms of sines and cosines.

\frac{{sin}^{2} (x)}{{cos}^{2} (x)} + \frac{2 sin (x)}{cos (x)} \cdot \frac{1}{cos (x)} + {sec}^{2} (x)

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

Step 2.6

Multiply

\frac{2 sin (x)}{cos (x)} \cdot \frac{1}{cos (x)}

$\frac{2 \sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}$ .

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Step 2.6.1

Multiply

\frac{2 sin (x)}{cos (x)}

$\frac{2 \sin (x)}{\cos (x)}$ by

\frac{1}{cos (x)}

$\frac{1}{\cos (x)}$ .

\frac{{sin}^{2} (x)}{{cos}^{2} (x)} + \frac{2 sin (x)}{cos (x) cos (x)} + {sec}^{2} (x)

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

Step 2.6.2

Raise

cos (x)

$\cos (x)$ to the power of

1

$1$ .

\frac{{sin}^{2} (x)}{{cos}^{2} (x)} + \frac{2 sin (x)}{{cos}^{1} (x) cos (x)} + {sec}^{2} (x)

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

Step 2.6.3

Raise

cos (x)

$\cos (x)$ to the power of

1

$1$ .

\frac{{sin}^{2} (x)}{{cos}^{2} (x)} + \frac{2 sin (x)}{{cos}^{1} (x) {cos}^{1} (x)} + {sec}^{2} (x)

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

Step 2.6.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{{sin}^{2} (x)}{{cos}^{2} (x)} + \frac{2 sin (x)}{{cos (x)}^{1 + 1}} + {sec}^{2} (x)

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

Step 2.6.5

Add

1

$1$ and

1

$1$ .

\frac{{sin}^{2} (x)}{{cos}^{2} (x)} + \frac{2 sin (x)}{{cos}^{2} (x)} + {sec}^{2} (x)

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

\frac{{sin}^{2} (x)}{{cos}^{2} (x)} + \frac{2 sin (x)}{{cos}^{2} (x)} + {sec}^{2} (x)

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

Step 2.7

Rewrite

sec (x)

$\sec (x)$ in terms of sines and cosines.

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

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

Step 2.8

Apply the product rule to

\frac{1}{cos (x)}

$\frac{1}{\cos (x)}$ .

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

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

Step 2.9

One to any power is one.

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

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

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

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

Step 3

Combine the numerators over the common denominator.

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

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

Step 4

Simplify the expression.

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Step 4.1

Factor

sin (x)

$\sin (x)$ out of

{sin}^{2} (x) + 2 sin (x)

$\sin^{2} (x) + 2 \sin (x)$ .

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Step 4.1.1

Factor

$\sin (x)$ out of

$\sin^{2} (x)$ .

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

Step 4.1.2

Factor

$\sin (x)$ out of

$2 \sin (x)$ .

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

Step 4.1.3

Factor

$\sin (x)$ out of

$\sin (x) \sin (x) + \sin (x) \cdot 2$ .

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

Step 4.2

Combine the numerators over the common denominator.

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

Step 4.3

Simplify the numerator.

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Step 4.3.1

Apply the distributive property.

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

Step 4.3.2

Multiply

$\sin (x) \sin (x)$ .

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Step 4.3.2.1

Raise

$\sin (x)$ to the power of

$1$ .

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

Step 4.3.2.2

Raise

$\sin (x)$ to the power of

$1$ .

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

Step 4.3.2.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.3.2.4

Add

$1$ and

$1$ .

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

Step 4.3.3

Move

$2$ to the left of

$\sin (x)$ .

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

Step 4.3.4

Rewrite

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

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Step 4.3.4.1

Let

$u = \sin (x)$ . Substitute

$u$ for all occurrences of

$\sin (x)$ .

$\frac{u^{2} + 2 u + 1}{\cos^{2} (x)}$

Step 4.3.4.2

Factor using the perfect square rule.

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Step 4.3.4.2.1

Rewrite

$1$ as

$1^{2}$ .

$\frac{u^{2} + 2 u + 1^{2}}{\cos^{2} (x)}$

Step 4.3.4.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 u = 2 \cdot u \cdot 1$

Step 4.3.4.2.3

Rewrite the polynomial.

$\frac{u^{2} + 2 \cdot u \cdot 1 + 1^{2}}{\cos^{2} (x)}$

Step 4.3.4.2.4

Factor using the perfect square trinomial rule

$a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where

$a = u$ and

$b = 1$ .

$\frac{{(u + 1)}^{2}}{\cos^{2} (x)}$

Step 4.3.4.3

Replace all occurrences of

$u$ with

$\sin (x)$ .

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

Step 5

Apply Pythagorean identity in reverse.

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

Step 6

Simplify the denominator.

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

Step 7

Cancel the common factor of

${(1 + \sin (x))}^{2}$ and

$1 + \sin (x)$ .

$\frac{1 + \sin (x)}{1 - \sin (x)}$

Step 8

Rewrite

$\frac{1 + \sin (x)}{1 - \sin (x)}$ as

$\frac{\frac{1}{\sin (x)} + 1}{\frac{1}{\sin (x)} - 1}$ .

$\frac{\frac{1}{\sin (x)} + 1}{\frac{1}{\sin (x)} - 1}$

Step 9

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

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