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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Name

Name	three hundred fifty-eight million five thousand four hundred sixty-eight

Interesting facts

358005468 has 32 divisors, whose sum is 1074487680
The reverse of 358005468 is 864500853
Previous prime number is 10151

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 39
Digital Root 3

Name

Name	two hundred five million eight hundred sixty-two thousand eight hundred forty-three

Interesting facts

205862843 has 4 divisors, whose sum is 217972440
The reverse of 205862843 is 348268502
Previous prime number is 17

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 38
Digital Root 2

Name

Name	eight hundred seventy-three million nine hundred sixty-eight thousand two hundred seventy-six

Interesting facts

873968276 has 16 divisors, whose sum is 1968808140
The reverse of 873968276 is 672869378
Previous prime number is 829

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 56
Digital Root 2

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