Trigonometry Examples

$\sin (t) \cos (t) (\tan (t) + \cot (t)) = 1$

Step 1

Start on the left side.

$\sin (t) \cos (t) (\tan (t) + \cot (t))$

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$\sin (t) \cos (t) (\frac{\sin (t)}{\cos (t)} + \cot (t))$

Step 2.1.2

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

$\sin (t) \cos (t) (\frac{\sin (t)}{\cos (t)} + \frac{\cos (t)}{\sin (t)})$

Step 2.2

Apply the distributive property.

$\sin (t) \cos (t) \frac{\sin (t)}{\cos (t)} + \sin (t) \cos (t) \frac{\cos (t)}{\sin (t)}$

Step 2.3

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

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Step 2.3.1

Factor $\cos (t)$ out of $\sin (t) \cos (t)$ .

$\cos (t) \sin (t) \frac{\sin (t)}{\cos (t)} + \sin (t) \cos (t) \frac{\cos (t)}{\sin (t)}$

Step 2.3.2

Cancel the common factor.

$\cos (t) \sin (t) \frac{\sin (t)}{\cos (t)} + \sin (t) \cos (t) \frac{\cos (t)}{\sin (t)}$

Step 2.3.3

Rewrite the expression.

$\sin (t) \sin (t) + \sin (t) \cos (t) \frac{\cos (t)}{\sin (t)}$

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Add $1$ and $1$ .

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

Step 2.8

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

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Step 2.8.1

Factor $\sin (t)$ out of $\sin (t) \cos (t)$ .

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

Step 2.8.2

Cancel the common factor.

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

Step 2.8.3

Rewrite the expression.

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

Add $1$ and $1$ .

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

Step 2.13

Apply pythagorean identity.

$1$

Step 3

Because the two sides have been shown to be equivalent, the equation is an identity.

$\sin (t) \cos (t) (\tan (t) + \cot (t)) = 1$ is an identity