What is the general solution for sin^2(a)=((2tan(a)))/((1+tan^2(a))) ?

The general solution for sin^2(a)=((2tan(a)))/((1+tan^2(a))) is a=2pin,a=pi+2pin,a=1.10714…+pin

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sin^2(a)=((2tan(a)))/((1+tan^2(a)))

sin^{2} (a) = \frac{( 2 tan ( a ) )}{( 1 + tan ^{2} ( a ) )}

a = 2 πn, a = π + 2 πn, a = 1.10714 \dots + πn

Solution steps

sin^{2} (a) = \frac{( 2 tan ( a ) )}{( 1 + tan ^{2} ( a ) )}

Subtract

\frac{2 tan ( a )}{1 + tan ^{2} ( a )}

from both sides

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

Simplify

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

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

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

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

Express with sin, cos

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

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

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

Factor

(cos^{2} (a) + sin^{2} (a)) sin^{2} (a) - 2 cos (a) sin (a) : sin (a) (sin (a) (cos^{2} (a) + sin^{2} (a)) - 2 cos (a))

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

Rewrite using trig identities

sin (a) (- 2 cos (a) + sin (a)) = 0

Solving each part separately

sin (a) = 0 or - 2 cos (a) + sin (a) = 0

sin (a) = 0 : a = 2 πn, a = π + 2 πn

- 2 cos (a) + sin (a) = 0 : a = arctan (2) + πn

Combine all the solutions

a = 2 πn, a = π + 2 πn, a = arctan (2) + πn

Show solutions in decimal form

a = 2 πn, a = π + 2 πn, a = 1.10714 \dots + πn

\frac{tan ( a )}{2} = \frac{2}{11}

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

cos (x) [3 sin (x) - 2] = 0

sin^{3} (x) + sin (x) - 4 = 0

a, sin^{2} (a) + cos^{2} (b) = 1

What is the general solution for sin^2(a)=((2tan(a)))/((1+tan^2(a))) ?
The general solution for sin^2(a)=((2tan(a)))/((1+tan^2(a))) is a=2pin,a=pi+2pin,a=1.10714…+pin