Precalculus Examples

$\sec^{2} (x) + \csc^{2} (x) = \sec^{2} (x) \cdot \csc^{2} (x)$

Step 1

Subtract $\sec^{2} (x) \cdot \csc^{2} (x)$ from both sides of the equation.

$\sec^{2} (x) + \csc^{2} (x) - \sec^{2} (x) \cdot \csc^{2} (x) = 0$

Step 2

Replace the $\sec^{2} (x)$ with $1 + \tan^{2} (x)$ based on the $\tan^{2} (x) + 1 = \sec^{2} (x)$ identity.

$(1 + \tan^{2} (x)) + \csc^{2} (x) - \sec^{2} (x) \cdot \csc^{2} (x) = 0$

Step 3

Multiply $- \sec^{2} (x)$ by $\csc^{2} (x)$ .

$1 + \tan^{2} (x) + \csc^{2} (x) - \sec^{2} (x) \csc^{2} (x) = 0$

Step 4

Simplify the left side.

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Step 4.1

Simplify $1 + \tan^{2} (x) + \csc^{2} (x) - \sec^{2} (x) \csc^{2} (x)$ .

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Step 4.1.1

Rearrange terms.

$\tan^{2} (x) + 1 + \csc^{2} (x) - \sec^{2} (x) \csc^{2} (x) = 0$

Step 4.1.2

Apply pythagorean identity.

$\sec^{2} (x) + \csc^{2} (x) - \sec^{2} (x) \csc^{2} (x) = 0$

Step 4.1.3

Simplify with factoring out.

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Step 4.1.3.1

Move $- \sec^{2} (x) \csc^{2} (x)$ .

$\sec^{2} (x) - \sec^{2} (x) \csc^{2} (x) + \csc^{2} (x) = 0$

Step 4.1.3.2

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

$- \sec^{2} (x) \cdot - 1 - \sec^{2} (x) \csc^{2} (x) + \csc^{2} (x) = 0$

Step 4.1.3.3

Factor $- \sec^{2} (x)$ out of $- \sec^{2} (x) \csc^{2} (x)$ .

$- \sec^{2} (x) \cdot - 1 - \sec^{2} (x) (\csc^{2} (x)) + \csc^{2} (x) = 0$

Step 4.1.3.4

Factor $- \sec^{2} (x)$ out of $- \sec^{2} (x) \cdot - 1 - \sec^{2} (x) (\csc^{2} (x))$ .

$- \sec^{2} (x) \cdot (- 1 + \csc^{2} (x)) + \csc^{2} (x) = 0$

Step 4.1.4

Rearrange terms.

$- \sec^{2} (x) \cdot (\csc^{2} (x) - 1) + \csc^{2} (x) = 0$

Step 4.1.5

Apply pythagorean identity.

$- \sec^{2} (x) \cdot \cot^{2} (x) + \csc^{2} (x) = 0$

Step 4.1.6

Simplify the expression.

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Step 4.1.6.1

Rewrite $\sec^{2} (x) \cdot \cot^{2} (x)$ as ${(\sec (x) \cot (x))}^{2}$ .

$- {(\sec (x) \cot (x))}^{2} + \csc^{2} (x) = 0$

Step 4.1.6.2

Reorder $- {(\sec (x) \cot (x))}^{2}$ and $\csc^{2} (x)$ .

$\csc^{2} (x) - {(\sec (x) \cot (x))}^{2} = 0$

Step 4.1.7

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

$(\csc (x) + \sec (x) \cot (x)) (\csc (x) - (\sec (x) \cot (x))) = 0$

Step 4.1.8

Simplify terms.

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Step 4.1.8.1

Simplify each term.

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Step 4.1.8.1.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 4.1.8.1.1.1

Reorder $\sec (x)$ and $\cot (x)$ .

$(\csc (x) + \cot (x) \sec (x)) (\csc (x) - (\sec (x) \cot (x))) = 0$

Step 4.1.8.1.1.2

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

$(\csc (x) + \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos (x)}) (\csc (x) - (\sec (x) \cot (x))) = 0$

Step 4.1.8.1.1.3

Cancel the common factors.

$(\csc (x) + \frac{1}{\sin (x)}) (\csc (x) - (\sec (x) \cot (x))) = 0$

Step 4.1.8.1.2

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

$(\csc (x) + \csc (x)) (\csc (x) - (\sec (x) \cot (x))) = 0$

Step 4.1.8.2

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

$2 \csc (x) (\csc (x) - (\sec (x) \cot (x))) = 0$

Step 4.1.8.3

Simplify each term.

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Step 4.1.8.3.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 4.1.8.3.1.1

Reorder $\sec (x)$ and $\cot (x)$ .

$2 \csc (x) (\csc (x) - (\cot (x) \sec (x))) = 0$

Step 4.1.8.3.1.2

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

$2 \csc (x) (\csc (x) - (\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos (x)})) = 0$

Step 4.1.8.3.1.3

Cancel the common factors.

$2 \csc (x) (\csc (x) - \frac{1}{\sin (x)}) = 0$

Step 4.1.8.3.2

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

$2 \csc (x) (\csc (x) - \csc (x)) = 0$

Step 4.1.8.4

Subtract $\csc (x)$ from $\csc (x)$ .

$2 \csc (x) \cdot 0 = 0$

Step 4.1.9

Multiply $2 \csc (x) \cdot 0$ .

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Step 4.1.9.1

Multiply $0$ by $2$ .

$0 \csc (x) = 0$

Step 4.1.9.2

Multiply $0$ by $\csc (x)$ .

$0 = 0$

Step 5

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

Always true