Trigonometry Examples

$\sin (u) \csc (u) - \cos^{2} (u) = \sin^{2} (u)$

Step 1

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

$\sin (u) \csc (u) - \cos^{2} (u) - \sin^{2} (u) = 0$

Step 2

Simplify $\sin (u) \csc (u) - \cos^{2} (u) - \sin^{2} (u)$ .

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Step 2.1

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

$\sin (u) \csc (u) - (\cos^{2} (u)) - \sin^{2} (u) = 0$

Step 2.2

Factor $- 1$ out of $- \sin^{2} (u)$ .

$\sin (u) \csc (u) - (\cos^{2} (u)) - (\sin^{2} (u)) = 0$

Step 2.3

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

$\sin (u) \csc (u) - (\cos^{2} (u) + \sin^{2} (u)) = 0$

Step 2.4

Rearrange terms.

$\sin (u) \csc (u) - (\sin^{2} (u) + \cos^{2} (u)) = 0$

Step 2.5

Apply pythagorean identity.

$\sin (u) \csc (u) - 1 \cdot 1 = 0$

Step 2.6

Simplify each term.

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Step 2.6.1

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

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Step 2.6.1.1

Reorder $\sin (u)$ and $\csc (u)$ .

$\csc (u) \sin (u) - 1 \cdot 1 = 0$

Step 2.6.1.2

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

$\frac{1}{\sin (u)} \sin (u) - 1 \cdot 1 = 0$

Step 2.6.1.3

Cancel the common factors.

$1 - 1 \cdot 1 = 0$

Step 2.6.2

Multiply $- 1$ by $1$ .

$1 - 1 = 0$

Step 2.7

Subtract $1$ from $1$ .

$0 = 0$

Step 3

Since $0 = 0$ , the equation will always be true for any value of $u$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$