Trigonometry Examples

$4 \cot^{2} (2 x) = \sec^{2} (x) + \csc^{2} (x) - 4$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

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

$4 \cot^{2} (2 x) - \sec^{2} (x) = \csc^{2} (x) - 4$

Step 1.2

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

$4 \cot^{2} (2 x) - \sec^{2} (x) - \csc^{2} (x) = - 4$

Step 1.3

Add $4$ to both sides of the equation.

$4 \cot^{2} (2 x) - \sec^{2} (x) - \csc^{2} (x) + 4 = 0$

Step 2

Simplify $4 \cot^{2} (2 x) - \sec^{2} (x) - \csc^{2} (x) + 4$ .

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Step 2.1

Move $4$ .

$4 \cot^{2} (2 x) + 4 - \sec^{2} (x) - \csc^{2} (x) = 0$

Step 2.2

Reorder $4 \cot^{2} (2 x)$ and $4$ .

$4 + 4 \cot^{2} (2 x) - \sec^{2} (x) - \csc^{2} (x) = 0$

Step 2.3

Factor $4$ out of $4$ .

$4 (1) + 4 \cot^{2} (2 x) - \sec^{2} (x) - \csc^{2} (x) = 0$

Step 2.4

Factor $4$ out of $4 \cot^{2} (2 x)$ .

$4 (1) + 4 (\cot^{2} (2 x)) - \sec^{2} (x) - \csc^{2} (x) = 0$

Step 2.5

Factor $4$ out of $4 (1) + 4 (\cot^{2} (2 x))$ .

$4 (1 + \cot^{2} (2 x)) - \sec^{2} (x) - \csc^{2} (x) = 0$

Step 2.6

Apply pythagorean identity.

$4 \csc^{2} (2 x) - \sec^{2} (x) - \csc^{2} (x) = 0$

Step 3

Set the argument in $\csc (2 x)$ equal to $π n$ to find where the expression is undefined.

$2 x = π n$ , for any integer $n$

Step 4

Divide each term in $2 x = π n$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = π n$ by $2$ .

$\frac{2 x}{2} = \frac{π n}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π n}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{π n}{2}$

Step 5

Set the argument in $\sec (x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$x = \frac{π}{2} + π n$ , for any integer $n$

Step 6

Set the argument in $\csc (x)$ equal to $π n$ to find where the expression is undefined.

$x = π n$ , for any integer $n$

Step 7

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

${x | x = \frac{π n}{2}, x = \frac{π}{2} + π n, x = π n}$ $n$ , for any integer $n$

Step 8