Simplify (2sin(x)cos(x)+(sin(x)-cos(x))^2)/(sec(x))

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Trigonometry

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

Apply the sine double-angle identity.

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

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

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

Multiply the numerator by the reciprocal of the denominator.

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

Apply the distributive property.

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

Simplify each term.

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

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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Simplify each term.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

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

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Multiply $- 1$ by $- 1$ .

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

Apply pythagorean identity.

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

Apply the distributive property.

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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