What is the general solution for tan(2A)=((2tan(A)))/([1+(tan(A))^2]) ?

The general solution for tan(2A)=((2tan(A)))/([1+(tan(A))^2]) is A=pin

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tan(2A)=((2tan(A)))/([1+(tan(A))^2])

tan (2 A) = \frac{( 2 tan ( A ) )}{[ 1 + ( tan ( A ) ) ^{2} ]}

A = πn

Solution steps

tan (2 A) = \frac{( 2 tan ( A ) )}{[ 1 + ( tan ( A ) ) ^{2} ]}

Subtract

\frac{2 tan ( A )}{[ 1 + ( tan ( A ) ) ^{2} ]}

from both sides

tan (2 A) - \frac{2 tan ( A )}{1 + tan ^{2} ( A )} = 0

Simplify

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

\frac{tan ( 2 A ) ( 1 + tan ^{2} ( A ) ) - 2 tan ( A )}{1 + tan ^{2} ( A )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

tan (2 A) (1 + tan^{2} (A)) - 2 tan (A) = 0

Rewrite using trig identities

\frac{4 tan ^{3} ( A )}{1 - tan ^{2} ( A )} = 0

Solve by substitution

tan (A) = 0

tan (A) = 0 : A = πn

Combine all the solutions

A = πn

cos (4 x) + 1 = 3 sin (2 x)

tan (2 t + 15) = cot (6 t - 5)

3 sin (θ) = 0

2 cos^{3} (x) - cos^{2} (x) + 2 cos (x) - 1 = 0

x, - 44 sin (2 x) + 42 cos (2 x) = 0

What is the general solution for tan(2A)=((2tan(A)))/([1+(tan(A))^2]) ?
The general solution for tan(2A)=((2tan(A)))/([1+(tan(A))^2]) is A=pin