Trigonometry Examples

$\frac{\tan^{2} (t)}{\sec (t)} = \sec (t) - \cos (t)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\frac{\tan^{2} (t)}{\sec (t)}$ .

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Step 1.1.1

Separate fractions.

$\frac{\tan^{2} (t)}{1} \cdot \frac{1}{\sec (t)} = \sec (t) - \cos (t)$

Step 1.1.2

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

$\frac{\tan^{2} (t)}{1} \cdot \frac{1}{\frac{1}{\cos (t)}} = \sec (t) - \cos (t)$

Step 1.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (t)}$ .

$\frac{\tan^{2} (t)}{1} (1 \cos (t)) = \sec (t) - \cos (t)$

Step 1.1.4

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

$\frac{\tan^{2} (t)}{1} \cos (t) = \sec (t) - \cos (t)$

Step 1.1.5

Divide $\tan^{2} (t)$ by $1$ .

$\tan^{2} (t) \cos (t) = \sec (t) - \cos (t)$

Step 1.1.6

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

${(\frac{\sin (t)}{\cos (t)})}^{2} \cos (t) = \sec (t) - \cos (t)$

Step 1.1.7

Apply the product rule to $\frac{\sin (t)}{\cos (t)}$ .

$\frac{\sin^{2} (t)}{\cos^{2} (t)} \cos (t) = \sec (t) - \cos (t)$

Step 1.1.8

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

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Step 1.1.8.1

Factor $\cos (t)$ out of $\cos^{2} (t)$ .

$\frac{\sin^{2} (t)}{\cos (t) \cos (t)} \cos (t) = \sec (t) - \cos (t)$

Step 1.1.8.2

Cancel the common factor.

$\frac{\sin^{2} (t)}{\cos (t) \cos (t)} \cos (t) = \sec (t) - \cos (t)$

Step 1.1.8.3

Rewrite the expression.

$\frac{\sin^{2} (t)}{\cos (t)} = \sec (t) - \cos (t)$

Step 2

Simplify the right side.

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Step 2.1

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

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

Step 3

Multiply both sides of the equation by $\cos (t)$ .

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Apply the distributive property.

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

Rewrite using the commutative property of multiplication.

$\sin^{2} (t) = 1 - \cos (t) \cos (t)$

Step 8

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

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Step 8.1

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

$\sin^{2} (t) = 1 - (\cos^{1} (t) \cos (t))$

Step 8.2

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

$\sin^{2} (t) = 1 - (\cos^{1} (t) \cos^{1} (t))$

Step 8.3

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

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

Step 8.4

Add $1$ and $1$ .

$\sin^{2} (t) = 1 - \cos^{2} (t)$

Step 9

Apply pythagorean identity.

$\sin^{2} (t) = \sin^{2} (t)$

Step 10

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| \sin (t) | = | \sin (t) |$

Step 11

Solve for $t$ .

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Step 11.1

Rewrite the absolute value equation as four equations without absolute value bars.

$\sin (t) = \sin (t)$

$\sin (t) = - \sin (t)$

$- \sin (t) = \sin (t)$

$- \sin (t) = - \sin (t)$

Step 11.2

After simplifying, there are only two unique equations to be solved.

$\sin (t) = \sin (t)$

$\sin (t) = - \sin (t)$

Step 11.3

Solve $\sin (t) = \sin (t)$ for $t$ .

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Step 11.3.1

For the two functions to be equal, the arguments of each must be equal.

$t = t$

Step 11.3.2

Move all terms containing $t$ to the left side of the equation.

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Step 11.3.2.1

Subtract $t$ from both sides of the equation.

$t - t = 0$

Step 11.3.2.2

Subtract $t$ from $t$ .

$0 = 0$

Step 11.3.3

Since $0 = 0$ , the equation will always be true.

All real numbers

Step 11.4

Solve $\sin (t) = - \sin (t)$ for $t$ .

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Step 11.4.1

Move all terms containing $\sin (t)$ to the left side of the equation.

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Step 11.4.1.1

Add $\sin (t)$ to both sides of the equation.

$\sin (t) + \sin (t) = 0$

Step 11.4.1.2

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

$2 \sin (t) = 0$

Step 11.4.2

Divide each term in $2 \sin (t) = 0$ by $2$ and simplify.

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Step 11.4.2.1

Divide each term in $2 \sin (t) = 0$ by $2$ .

$\frac{2 \sin (t)}{2} = \frac{0}{2}$

Step 11.4.2.2

Simplify the left side.

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Step 11.4.2.2.1

Cancel the common factor of $2$ .

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Step 11.4.2.2.1.1

Cancel the common factor.

$\frac{2 \sin (t)}{2} = \frac{0}{2}$

Step 11.4.2.2.1.2

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

$\sin (t) = \frac{0}{2}$

Step 11.4.2.3

Simplify the right side.

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Step 11.4.2.3.1

Divide $0$ by $2$ .

$\sin (t) = 0$

Step 11.4.3

Take the inverse sine of both sides of the equation to extract $t$ from inside the sine.

$t = \arcsin (0)$

Step 11.4.4

Simplify the right side.

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Step 11.4.4.1

The exact value of $\arcsin (0)$ is $0$ .

$t = 0$

Step 11.4.5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$t = π - 0$

Step 11.4.6

Subtract $0$ from $π$ .

$t = π$

Step 11.4.7

Find the period of $\sin (t)$ .

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Step 11.4.7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 11.4.7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 11.4.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 11.4.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 11.4.8

The period of the $\sin (t)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$t = 2 π n, π + 2 π n$ , for any integer $n$

Step 12

Consolidate the answers.

$t = π n$ , for any integer $n$