Trigonometry Examples

$\frac{2 \sin (x) \cos (x) + {(\sin (x) - \cos (x))}^{2}}{\sec (x)}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the sine double-angle identity.

$\frac{\sin (2 x) + {(\sin (x) - \cos (x))}^{2}}{\sec (x)}$

Step 1.2

Rewrite ${(\sin (x) - \cos (x))}^{2}$ as $(\sin (x) - \cos (x)) (\sin (x) - \cos (x))$ .

$\frac{\sin (2 x) + (\sin (x) - \cos (x)) (\sin (x) - \cos (x))}{\sec (x)}$

Step 1.3

Expand $(\sin (x) - \cos (x)) (\sin (x) - \cos (x))$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$\frac{\sin (2 x) + \sin (x) (\sin (x) - \cos (x)) - \cos (x) (\sin (x) - \cos (x))}{\sec (x)}$

Step 1.3.2

Apply the distributive property.

$\frac{\sin (2 x) + \sin (x) \sin (x) + \sin (x) (- \cos (x)) - \cos (x) (\sin (x) - \cos (x))}{\sec (x)}$

Step 1.3.3

Apply the distributive property.

$\frac{\sin (2 x) + \sin (x) \sin (x) + \sin (x) (- \cos (x)) - \cos (x) \sin (x) - \cos (x) (- \cos (x))}{\sec (x)}$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $\sin (x) \sin (x)$ .

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Step 1.4.1.1.1

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

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

Step 1.4.1.1.2

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

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

Step 1.4.1.1.3

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

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

Step 1.4.1.1.4

Add $1$ and $1$ .

$\frac{\sin (2 x) + \sin^{2} (x) + \sin (x) (- \cos (x)) - \cos (x) \sin (x) - \cos (x) (- \cos (x))}{\sec (x)}$

Step 1.4.1.2

Rewrite using the commutative property of multiplication.

$\frac{\sin (2 x) + \sin^{2} (x) - \sin (x) \cos (x) - \cos (x) \sin (x) - \cos (x) (- \cos (x))}{\sec (x)}$

Step 1.4.1.3

Multiply $- \cos (x) (- \cos (x))$ .

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Step 1.4.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.4.1.3.2

Multiply $\cos (x)$ by $1$ .

$\frac{\sin (2 x) + \sin^{2} (x) - \sin (x) \cos (x) - \cos (x) \sin (x) + \cos (x) \cos (x)}{\sec (x)}$

Step 1.4.1.3.3

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

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

Step 1.4.1.3.4

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

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

Step 1.4.1.3.5

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

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

Step 1.4.1.3.6

Add $1$ and $1$ .

$\frac{\sin (2 x) + \sin^{2} (x) - \sin (x) \cos (x) - \cos (x) \sin (x) + \cos^{2} (x)}{\sec (x)}$

Step 1.4.2

Reorder the factors of $- \sin (x) \cos (x)$ .

$\frac{\sin (2 x) + \sin^{2} (x) - \cos (x) \sin (x) - \cos (x) \sin (x) + \cos^{2} (x)}{\sec (x)}$

Step 1.4.3

Subtract $\cos (x) \sin (x)$ from $- \cos (x) \sin (x)$ .

$\frac{\sin (2 x) + \sin^{2} (x) - 2 \cos (x) \sin (x) + \cos^{2} (x)}{\sec (x)}$

Step 1.5

Move $\cos^{2} (x)$ .

$\frac{\sin (2 x) + \sin^{2} (x) + \cos^{2} (x) - 2 \cos (x) \sin (x)}{\sec (x)}$

Step 1.6

Apply pythagorean identity.

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

Step 2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Apply the distributive property.

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

Step 5

Simplify.

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Step 5.1

Multiply $\cos (x)$ by $1$ .

$\sin (2 x) \cos (x) + \cos (x) - 2 \cos (x) \sin (x) \cos (x)$

Step 5.2

Multiply $- 2 \cos (x) \sin (x) \cos (x)$ .

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

Add $1$ and $1$ .

$\sin (2 x) \cos (x) + \cos (x) - 2 \cos^{2} (x) \sin (x)$