Evaluate tan(2A)=(2tan(A))/(1-tan(A)^2)

$\tan (2 A) = \frac{2 \tan (A)}{1 - \tan^{2} (A)}$

Multiply both sides by $1 - \tan^{2} (A)$ .

$\tan (2 A) (1 - \tan^{2} (A)) = \frac{2 \tan (A)}{1 - \tan^{2} (A)} (1 - \tan^{2} (A))$

Simplify.

Simplify the left side.

Simplify $\tan (2 A) (1 - \tan^{2} (A))$ .

Apply the distributive property.

$\tan (2 A) \cdot 1 + \tan (2 A) (- \tan^{2} (A)) = \frac{2 \tan (A)}{1 - \tan^{2} (A)} (1 - \tan^{2} (A))$

Simplify the expression.

Multiply $\tan (2 A)$ by $1$ .

$\tan (2 A) + \tan (2 A) (- \tan^{2} (A)) = \frac{2 \tan (A)}{1 - \tan^{2} (A)} (1 - \tan^{2} (A))$

Rewrite using the commutative property of multiplication.

$\tan (2 A) - \tan (2 A) \tan^{2} (A) = \frac{2 \tan (A)}{1 - \tan^{2} (A)} (1 - \tan^{2} (A))$

Simplify the right side.

Cancel the common factor of $1 - \tan^{2} (A)$ .

Cancel the common factor.

$\tan (2 A) - \tan (2 A) \tan^{2} (A) = \frac{2 \tan (A)}{1 - \tan^{2} (A)} (1 - \tan^{2} (A))$

Rewrite the expression.

$\tan (2 A) - \tan (2 A) \tan^{2} (A) = 2 \tan (A)$

Solve for $A$ .

Simplify the left side.

Simplify each term.

Rewrite $\tan (2 A)$ in terms of sines and cosines.

$\frac{\sin (2 A)}{\cos (2 A)} - \tan (2 A) \tan^{2} (A) = 2 \tan (A)$

Rewrite $\tan (2 A)$ in terms of sines and cosines.

$\frac{\sin (2 A)}{\cos (2 A)} - \frac{\sin (2 A)}{\cos (2 A)} \tan^{2} (A) = 2 \tan (A)$

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

$\frac{\sin (2 A)}{\cos (2 A)} - \frac{\sin (2 A)}{\cos (2 A)} {(\frac{\sin (A)}{\cos (A)})}^{2} = 2 \tan (A)$

Apply the product rule to $\frac{\sin (A)}{\cos (A)}$ .

$\frac{\sin (2 A)}{\cos (2 A)} - \frac{\sin (2 A)}{\cos (2 A)} \cdot \frac{\sin^{2} (A)}{\cos^{2} (A)} = 2 \tan (A)$

Multiply $\frac{\sin^{2} (A)}{\cos^{2} (A)}$ by $\frac{\sin (2 A)}{\cos (2 A)}$ .

$\frac{\sin (2 A)}{\cos (2 A)} - \frac{\sin^{2} (A) \sin (2 A)}{\cos^{2} (A) \cos (2 A)} = 2 \tan (A)$

Simplify the numerator.

Apply the sine double-angle identity.

$\frac{\sin (2 A)}{\cos (2 A)} - \frac{\sin^{2} (A) \cdot 2 \sin (A) \cos (A)}{\cos^{2} (A) \cos (2 A)} = 2 \tan (A)$

Combine exponents.

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

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

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

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

Add $2$ and $1$ .

$\frac{\sin (2 A)}{\cos (2 A)} - \frac{2 \sin^{3} (A) \cos (A)}{\cos^{2} (A) \cos (2 A)} = 2 \tan (A)$

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

$\frac{\sin (2 A)}{\cos (2 A)} - \frac{2 \sin^{3} (A) \cos (A)}{\cos^{2} (A) (2 \cos^{2} (A) - 1)} = 2 \tan (A)$

Cancel the common factor of $\cos (A)$ and $\cos^{2} (A)$ .

Factor $\cos (A)$ out of $2 \sin^{3} (A) \cos (A)$ .

$\frac{\sin (2 A)}{\cos (2 A)} - \frac{\cos (A) (2 \sin^{3} (A))}{\cos^{2} (A) (2 \cos^{2} (A) - 1)} = 2 \tan (A)$

Cancel the common factors.

Factor $\cos (A)$ out of $\cos^{2} (A) (2 \cos^{2} (A) - 1)$ .

$\frac{\sin (2 A)}{\cos (2 A)} - \frac{\cos (A) (2 \sin^{3} (A))}{\cos (A) (\cos (A) (2 \cos^{2} (A) - 1))} = 2 \tan (A)$

Cancel the common factor.

$\frac{\sin (2 A)}{\cos (2 A)} - \frac{\cos (A) (2 \sin^{3} (A))}{\cos (A) (\cos (A) (2 \cos^{2} (A) - 1))} = 2 \tan (A)$

Rewrite the expression.

$\frac{\sin (2 A)}{\cos (2 A)} - \frac{2 \sin^{3} (A)}{\cos (A) (2 \cos^{2} (A) - 1)} = 2 \tan (A)$

Apply the cosine double-angle identity.

$\frac{\sin (2 A)}{\cos (2 A)} - \frac{2 \sin^{3} (A)}{\cos (A) \cos (2 A)} = 2 \tan (A)$

Simplify the right side.

Simplify $2 \tan (A)$ .

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

$\frac{\sin (2 A)}{\cos (2 A)} - \frac{2 \sin^{3} (A)}{\cos (A) \cos (2 A)} = 2 \frac{\sin (A)}{\cos (A)}$

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

$\frac{\sin (2 A)}{\cos (2 A)} - \frac{2 \sin^{3} (A)}{\cos (A) \cos (2 A)} = \frac{2 \sin (A)}{\cos (A)}$

Multiply both sides of the equation by $\cos (2 A)$ .

$\cos (2 A) (\frac{\sin (2 A)}{\cos (2 A)} - \frac{2 \sin^{3} (A)}{\cos (A) \cos (2 A)}) = \cos (2 A) \frac{2 \sin (A)}{\cos (A)}$

Apply the distributive property.

$\cos (2 A) \frac{\sin (2 A)}{\cos (2 A)} + \cos (2 A) (- \frac{2 \sin^{3} (A)}{\cos (A) \cos (2 A)}) = \cos (2 A) \frac{2 \sin (A)}{\cos (A)}$

Cancel the common factor of $\cos (2 A)$ .

Cancel the common factor.

$\cos (2 A) \frac{\sin (2 A)}{\cos (2 A)} + \cos (2 A) (- \frac{2 \sin^{3} (A)}{\cos (A) \cos (2 A)}) = \cos (2 A) \frac{2 \sin (A)}{\cos (A)}$

Rewrite the expression.

$\sin (2 A) + \cos (2 A) (- \frac{2 \sin^{3} (A)}{\cos (A) \cos (2 A)}) = \cos (2 A) \frac{2 \sin (A)}{\cos (A)}$

Rewrite using the commutative property of multiplication.

$\sin (2 A) - \cos (2 A) \frac{2 \sin^{3} (A)}{\cos (A) \cos (2 A)} = \cos (2 A) \frac{2 \sin (A)}{\cos (A)}$

Cancel the common factor of $\cos (2 A)$ .

Factor $\cos (2 A)$ out of $- \cos (2 A)$ .

$\sin (2 A) + \cos (2 A) \cdot - 1 \frac{2 \sin^{3} (A)}{\cos (A) \cos (2 A)} = \cos (2 A) \frac{2 \sin (A)}{\cos (A)}$

Factor $\cos (2 A)$ out of $\cos (A) \cos (2 A)$ .

$\sin (2 A) + \cos (2 A) \cdot - 1 \frac{2 \sin^{3} (A)}{\cos (2 A) \cos (A)} = \cos (2 A) \frac{2 \sin (A)}{\cos (A)}$

Cancel the common factor.

$\sin (2 A) + \cos (2 A) \cdot - 1 \frac{2 \sin^{3} (A)}{\cos (2 A) \cos (A)} = \cos (2 A) \frac{2 \sin (A)}{\cos (A)}$

Rewrite the expression.

$\sin (2 A) - \frac{2 \sin^{3} (A)}{\cos (A)} = \cos (2 A) \frac{2 \sin (A)}{\cos (A)}$

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

$\sin (2 A) - \frac{2 \sin^{3} (A)}{\cos (A)} = \frac{\cos (2 A) (2 \sin (A))}{\cos (A)}$

Move $2$ to the left of $\cos (2 A)$ .

$\sin (2 A) - \frac{2 \sin^{3} (A)}{\cos (A)} = \frac{2 \cos (2 A) \sin (A)}{\cos (A)}$

Multiply both sides of the equation by $\cos (A)$ .

$\cos (A) (\sin (2 A) - \frac{2 \sin^{3} (A)}{\cos (A)}) = \cos (A) \frac{2 \cos (2 A) \sin (A)}{\cos (A)}$

Apply the distributive property.

$\cos (A) \sin (2 A) + \cos (A) (- \frac{2 \sin^{3} (A)}{\cos (A)}) = \cos (A) \frac{2 \cos (2 A) \sin (A)}{\cos (A)}$

Rewrite using the commutative property of multiplication.

$\cos (A) \sin (2 A) - \cos (A) \frac{2 \sin^{3} (A)}{\cos (A)} = \cos (A) \frac{2 \cos (2 A) \sin (A)}{\cos (A)}$

Simplify each term.

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

Factor $\cos (A)$ out of $- \cos (A)$ .

$\cos (A) \sin (2 A) + \cos (A) \cdot - 1 \frac{2 \sin^{3} (A)}{\cos (A)} = \cos (A) \frac{2 \cos (2 A) \sin (A)}{\cos (A)}$

Cancel the common factor.

$\cos (A) \sin (2 A) + \cos (A) \cdot - 1 \frac{2 \sin^{3} (A)}{\cos (A)} = \cos (A) \frac{2 \cos (2 A) \sin (A)}{\cos (A)}$

Rewrite the expression.

$\cos (A) \sin (2 A) - (2 \sin^{3} (A)) = \cos (A) \frac{2 \cos (2 A) \sin (A)}{\cos (A)}$

Multiply $2$ by $- 1$ .

$\cos (A) \sin (2 A) - 2 \sin^{3} (A) = \cos (A) \frac{2 \cos (2 A) \sin (A)}{\cos (A)}$

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

Cancel the common factor.

$\cos (A) \sin (2 A) - 2 \sin^{3} (A) = \cos (A) \frac{2 \cos (2 A) \sin (A)}{\cos (A)}$

Rewrite the expression.

$\cos (A) \sin (2 A) - 2 \sin^{3} (A) = 2 \cos (2 A) \sin (A)$

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

$\cos (A) \sin (2 A) - 2 \sin^{3} (A) = 2 (1 - 2 \sin^{2} (A)) \sin (A)$

Subtract $2 (1 - 2 \sin^{2} (A)) \sin (A)$ from both sides of the equation.

$\cos (A) \sin (2 A) - 2 \sin^{3} (A) - 2 (1 - 2 \sin^{2} (A)) \sin (A) = 0$

Simplify the left side.

Simplify $\cos (A) \sin (2 A) - 2 \sin^{3} (A) - 2 (1 - 2 \sin^{2} (A)) \sin (A)$ .

Simplify terms.

Simplify each term.

Apply the sine double-angle identity.

$\cos (A) (2 \sin (A) \cos (A)) - 2 \sin^{3} (A) - 2 (1 - 2 \sin^{2} (A)) \sin (A) = 0$

Rewrite using the commutative property of multiplication.

$2 \cos (A) \sin (A) \cos (A) - 2 \sin^{3} (A) - 2 (1 - 2 \sin^{2} (A)) \sin (A) = 0$

Multiply $2 \cos (A) \sin (A) \cos (A)$ .

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

$2 (\cos^{1} (A) \cos (A)) \sin (A) - 2 \sin^{3} (A) - 2 (1 - 2 \sin^{2} (A)) \sin (A) = 0$

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

$2 (\cos^{1} (A) \cos^{1} (A)) \sin (A) - 2 \sin^{3} (A) - 2 (1 - 2 \sin^{2} (A)) \sin (A) = 0$

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

$2 {\cos (A)}^{1 + 1} \sin (A) - 2 \sin^{3} (A) - 2 (1 - 2 \sin^{2} (A)) \sin (A) = 0$

Add $1$ and $1$ .

$2 \cos^{2} (A) \sin (A) - 2 \sin^{3} (A) - 2 (1 - 2 \sin^{2} (A)) \sin (A) = 0$

Apply the distributive property.

$2 \cos^{2} (A) \sin (A) - 2 \sin^{3} (A) + (- 2 \cdot 1 - 2 (- 2 \sin^{2} (A))) \sin (A) = 0$

Multiply $- 2$ by $1$ .

$2 \cos^{2} (A) \sin (A) - 2 \sin^{3} (A) + (- 2 - 2 (- 2 \sin^{2} (A))) \sin (A) = 0$

Multiply $- 2$ by $- 2$ .

$2 \cos^{2} (A) \sin (A) - 2 \sin^{3} (A) + (- 2 + 4 \sin^{2} (A)) \sin (A) = 0$

Apply the distributive property.

$2 \cos^{2} (A) \sin (A) - 2 \sin^{3} (A) - 2 \sin (A) + 4 \sin^{2} (A) \sin (A) = 0$

Multiply $\sin^{2} (A)$ by $\sin (A)$ by adding the exponents.

Move $\sin (A)$ .

$2 \cos^{2} (A) \sin (A) - 2 \sin^{3} (A) - 2 \sin (A) + 4 (\sin (A) \sin^{2} (A)) = 0$

Multiply $\sin (A)$ by $\sin^{2} (A)$ .

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

$2 \cos^{2} (A) \sin (A) - 2 \sin^{3} (A) - 2 \sin (A) + 4 (\sin^{1} (A) \sin^{2} (A)) = 0$

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

$2 \cos^{2} (A) \sin (A) - 2 \sin^{3} (A) - 2 \sin (A) + 4 {\sin (A)}^{1 + 2} = 0$

Add $1$ and $2$ .

$2 \cos^{2} (A) \sin (A) - 2 \sin^{3} (A) - 2 \sin (A) + 4 \sin^{3} (A) = 0$

Simplify with factoring out.

Factor $2 \sin (A)$ out of $2 \cos^{2} (A) \sin (A) - 2 \sin^{3} (A) - 2 \sin (A) + 4 \sin^{3} (A)$ .

Factor $2 \sin (A)$ out of $2 \cos^{2} (A) \sin (A)$ .

$2 \sin (A) (\cos^{2} (A)) - 2 \sin^{3} (A) - 2 \sin (A) + 4 \sin^{3} (A) = 0$

Factor $2 \sin (A)$ out of $- 2 \sin^{3} (A)$ .

$2 \sin (A) (\cos^{2} (A)) + 2 \sin (A) (- \sin^{2} (A)) - 2 \sin (A) + 4 \sin^{3} (A) = 0$

Factor $2 \sin (A)$ out of $- 2 \sin (A)$ .

$2 \sin (A) (\cos^{2} (A)) + 2 \sin (A) (- \sin^{2} (A)) + 2 \sin (A) (- 1) + 4 \sin^{3} (A) = 0$

Factor $2 \sin (A)$ out of $4 \sin^{3} (A)$ .

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

Factor $2 \sin (A)$ out of $2 \sin (A) (\cos^{2} (A)) + 2 \sin (A) (- \sin^{2} (A))$ .

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

Factor $2 \sin (A)$ out of $2 \sin (A) (\cos^{2} (A) - \sin^{2} (A)) + 2 \sin (A) (- 1)$ .

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

Factor $2 \sin (A)$ out of $2 \sin (A) (\cos^{2} (A) - \sin^{2} (A) - 1) + 2 \sin (A) (2 \sin^{2} (A))$ .

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

Simplify the expression.

Move $- 1$ .

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

Reorder $\cos^{2} (A)$ and $- 1$ .

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

Rewrite $- 1$ as $- 1 (1)$ .

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

Factor $- 1$ out of $\cos^{2} (A)$ .

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

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (A))$ .

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

Rewrite $- 1 (1 - \cos^{2} (A))$ as $- (1 - \cos^{2} (A))$ .

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

Apply pythagorean identity.

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

Simplify by adding terms.

Subtract $\sin^{2} (A)$ from $- \sin^{2} (A)$ .

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

Add $- 2 \sin^{2} (A)$ and $2 \sin^{2} (A)$ .

$2 \sin (A) \cdot 0 = 0$

Multiply $2 \sin (A) \cdot 0$ .

Multiply $0$ by $2$ .

$0 \sin (A) = 0$

Multiply $0$ by $\sin (A)$ .

$0 = 0$

Since $0 = 0$ , the equation will always be true.

Always true

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$

Evaluate tan(2A)=(2tan(A))/(1-tan(A)^2)

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