Trigonometry Examples

$(1 - \tan (x)) (1 + \sin (2 x)) = 1 + \tan (x)$

Step 1

Simplify $(1 - \tan (x)) (1 + \sin (2 x))$ .

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Step 1.1

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

$(1 - \frac{\sin (x)}{\cos (x)}) (1 + \sin (2 x)) = 1 + \tan (x)$

Step 1.2

Expand $(1 - \frac{\sin (x)}{\cos (x)}) (1 + \sin (2 x))$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$1 (1 + \sin (2 x)) - \frac{\sin (x)}{\cos (x)} (1 + \sin (2 x)) = 1 + \tan (x)$

Step 1.2.2

Apply the distributive property.

$1 \cdot 1 + 1 \sin (2 x) - \frac{\sin (x)}{\cos (x)} (1 + \sin (2 x)) = 1 + \tan (x)$

Step 1.2.3

Apply the distributive property.

$1 \cdot 1 + 1 \sin (2 x) - \frac{\sin (x)}{\cos (x)} \cdot 1 - \frac{\sin (x)}{\cos (x)} \sin (2 x) = 1 + \tan (x)$

Step 1.3

Simplify terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $1$ by $1$ .

$1 + 1 \sin (2 x) - \frac{\sin (x)}{\cos (x)} \cdot 1 - \frac{\sin (x)}{\cos (x)} \sin (2 x) = 1 + \tan (x)$

Step 1.3.1.2

Multiply $\sin (2 x)$ by $1$ .

$1 + \sin (2 x) - \frac{\sin (x)}{\cos (x)} \cdot 1 - \frac{\sin (x)}{\cos (x)} \sin (2 x) = 1 + \tan (x)$

Step 1.3.1.3

Multiply $- 1$ by $1$ .

$1 + \sin (2 x) - \frac{\sin (x)}{\cos (x)} - \frac{\sin (x)}{\cos (x)} \sin (2 x) = 1 + \tan (x)$

Step 1.3.1.4

Combine $\sin (2 x)$ and $\frac{\sin (x)}{\cos (x)}$ .

$1 + \sin (2 x) - \frac{\sin (x)}{\cos (x)} - \frac{\sin (2 x) \sin (x)}{\cos (x)} = 1 + \tan (x)$

Step 1.3.1.5

Simplify the numerator.

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Step 1.3.1.5.1

Apply the sine double-angle identity.

$1 + \sin (2 x) - \frac{\sin (x)}{\cos (x)} - \frac{2 \sin (x) \cos (x) \sin (x)}{\cos (x)} = 1 + \tan (x)$

Step 1.3.1.5.2

Combine exponents.

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Step 1.3.1.5.2.1

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

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

Step 1.3.1.5.2.2

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

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

Step 1.3.1.5.2.3

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

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

Step 1.3.1.5.2.4

Add $1$ and $1$ .

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

Step 1.3.1.6

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

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Step 1.3.1.6.1

Cancel the common factor.

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

Step 1.3.1.6.2

Divide $2 \sin^{2} (x)$ by $1$ .

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

Step 1.3.1.7

Multiply $2$ by $- 1$ .

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

Step 1.3.2

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

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

Step 1.4

Apply the cosine double-angle identity.

$\cos (2 x) + \sin (2 x) - \frac{\sin (x)}{\cos (x)} = 1 + \tan (x)$

Step 1.5

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\cos (2 x) + \sin (2 x) - \tan (x) = 1 + \tan (x)$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 3