Calculus Examples

$\cos (a) (1 - \cos (2 a)) = \sin (a) \sin (2 a)$

Step 1

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

$\cos (a) (1 - (1 - 2 \sin^{2} (a))) = \sin (a) \sin (2 a)$

Step 2

Subtract $\sin (a) \sin (2 a)$ from both sides of the equation.

$\cos (a) (1 - (1 - 2 \sin^{2} (a))) - \sin (a) \sin (2 a) = 0$

Step 3

Simplify the left side.

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Step 3.1

Simplify $\cos (a) (1 - (1 - 2 \sin^{2} (a))) - \sin (a) \sin (2 a)$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Simplify each term.

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Step 3.1.1.1.1

Apply the distributive property.

$\cos (a) (1 - 1 \cdot 1 - (- 2 \sin^{2} (a))) - \sin (a) \sin (2 a) = 0$

Step 3.1.1.1.2

Multiply $- 1$ by $1$ .

$\cos (a) (1 - 1 - (- 2 \sin^{2} (a))) - \sin (a) \sin (2 a) = 0$

Step 3.1.1.1.3

Multiply $- 2$ by $- 1$ .

$\cos (a) (1 - 1 + 2 \sin^{2} (a)) - \sin (a) \sin (2 a) = 0$

Step 3.1.1.2

Subtract $1$ from $1$ .

$\cos (a) (0 + 2 \sin^{2} (a)) - \sin (a) \sin (2 a) = 0$

Step 3.1.1.3

Add $0$ and $2 \sin^{2} (a)$ .

$\cos (a) (2 \sin^{2} (a)) - \sin (a) \sin (2 a) = 0$

Step 3.1.1.4

Rewrite using the commutative property of multiplication.

$2 \cos (a) \sin^{2} (a) - \sin (a) \sin (2 a) = 0$

Step 3.1.1.5

Apply the sine double-angle identity.

$2 \cos (a) \sin^{2} (a) - \sin (a) (2 \sin (a) \cos (a)) = 0$

Step 3.1.1.6

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

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Step 3.1.1.6.1

Multiply $2$ by $- 1$ .

$2 \cos (a) \sin^{2} (a) - 2 \sin (a) (\sin (a) \cos (a)) = 0$

Step 3.1.1.6.2

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

$2 \cos (a) \sin^{2} (a) - 2 (\sin^{1} (a) \sin (a)) \cos (a) = 0$

Step 3.1.1.6.3

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

$2 \cos (a) \sin^{2} (a) - 2 (\sin^{1} (a) \sin^{1} (a)) \cos (a) = 0$

Step 3.1.1.6.4

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

$2 \cos (a) \sin^{2} (a) - 2 {\sin (a)}^{1 + 1} \cos (a) = 0$

Step 3.1.1.6.5

Add $1$ and $1$ .

$2 \cos (a) \sin^{2} (a) - 2 \sin^{2} (a) \cos (a) = 0$

Step 3.1.2

Combine the opposite terms in $2 \cos (a) \sin^{2} (a) - 2 \sin^{2} (a) \cos (a)$ .

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Step 3.1.2.1

Reorder the factors in the terms $2 \cos (a) \sin^{2} (a)$ and $- 2 \sin^{2} (a) \cos (a)$ .

$2 \sin^{2} (a) \cos (a) - 2 \sin^{2} (a) \cos (a) = 0$

Step 3.1.2.2

Subtract $2 \sin^{2} (a) \cos (a)$ from $2 \sin^{2} (a) \cos (a)$ .

$0 = 0$

Step 4

Since $0 = 0$ , the equation will always be true for any value of $a$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$