Trigonometry Examples

$\cos^{2} (x) + \csc^{2} (x) - 1$

Step 1

Simplify with factoring out.

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Step 1.1

Move $- 1$ .

$\cos^{2} (x) - 1 + \csc^{2} (x)$

Step 1.2

Reorder $\cos^{2} (x)$ and $- 1$ .

$- 1 + \cos^{2} (x) + \csc^{2} (x)$

Step 1.3

Rewrite $- 1$ as $- 1 (1)$ .

$- 1 (1) + \cos^{2} (x) + \csc^{2} (x)$

Step 1.4

Factor $- 1$ out of $\cos^{2} (x)$ .

$- 1 (1) - 1 (- \cos^{2} (x)) + \csc^{2} (x)$

Step 1.5

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (x))$ .

$- 1 (1 - \cos^{2} (x)) + \csc^{2} (x)$

Step 1.6

Rewrite $- 1 (1 - \cos^{2} (x))$ as $- (1 - \cos^{2} (x))$ .

$- (1 - \cos^{2} (x)) + \csc^{2} (x)$

Step 2

Apply pythagorean identity.

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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

Step 3.1.2

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 3.1.3

One to any power is one.

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

Step 3.2

Rewrite $1$ as $1^{2}$ .

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

Step 4

Rewrite $\frac{1^{2}}{\sin^{2} (x)}$ as ${(\frac{1}{\sin (x)})}^{2}$ .

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

Step 5

Reorder $- \sin^{2} (x)$ and ${(\frac{1}{\sin (x)})}^{2}$ .

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

Step 6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{1}{\sin (x)}$ and $b = \sin (x)$ .

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

Step 7

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

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

Step 8

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$(\csc (x) + \sin (x)) (\csc (x) - \sin (x))$

Step 9

Expand $(\csc (x) + \sin (x)) (\csc (x) - \sin (x))$ using the FOIL Method.

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Step 9.1

Apply the distributive property.

$\csc (x) (\csc (x) - \sin (x)) + \sin (x) (\csc (x) - \sin (x))$

Step 9.2

Apply the distributive property.

$\csc (x) \csc (x) + \csc (x) (- \sin (x)) + \sin (x) (\csc (x) - \sin (x))$

Step 9.3

Apply the distributive property.

$\csc (x) \csc (x) + \csc (x) (- \sin (x)) + \sin (x) \csc (x) + \sin (x) (- \sin (x))$

Step 10

Simplify terms.

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Step 10.1

Combine the opposite terms in $\csc (x) \csc (x) + \csc (x) (- \sin (x)) + \sin (x) \csc (x) + \sin (x) (- \sin (x))$ .

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Step 10.1.1

Reorder the factors in the terms $\csc (x) (- \sin (x))$ and $\sin (x) \csc (x)$ .

$\csc (x) \csc (x) - \csc (x) \sin (x) + \csc (x) \sin (x) + \sin (x) (- \sin (x))$

Step 10.1.2

Add $- \csc (x) \sin (x)$ and $\csc (x) \sin (x)$ .

$\csc (x) \csc (x) + 0 + \sin (x) (- \sin (x))$

Step 10.1.3

Add $\csc (x) \csc (x)$ and $0$ .

$\csc (x) \csc (x) + \sin (x) (- \sin (x))$

Step 10.2

Simplify each term.

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Step 10.2.1

Multiply $\csc (x) \csc (x)$ .

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Step 10.2.1.1

Raise $\csc (x)$ to the power of $1$ .

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

Step 10.2.1.2

Raise $\csc (x)$ to the power of $1$ .

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

Step 10.2.1.3

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

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

Step 10.2.1.4

Add $1$ and $1$ .

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

Step 10.2.2

Rewrite using the commutative property of multiplication.

$\csc^{2} (x) - \sin (x) \sin (x)$

Step 10.2.3

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

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Step 10.2.3.1

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

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

Step 10.2.3.2

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

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

Step 10.2.3.3

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

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

Step 10.2.3.4

Add $1$ and $1$ .

$\csc^{2} (x) - \sin^{2} (x)$