1 + sin²A = 3 sinA cos A cos²A + 2 sin²A = 3 sinA cosA (1= cos²A + sin² A) cotanA + 2 tanA = 3 ( all this * tan A) 1 + 2 tan²A =3 tanA 2 tan²A - 3 tanA +1 = 0 2 (tanA -1)*(tanA-1/2) =0 Then tan A = 1 or tan A =1/2...

Given:
1 sin^2a=3sinAcosA

To prove:
tanA=1 or 1/2

Proof:

Step 1:
Rewrite the given equation using trigonometric identities.

sin^2a = 3sinAcosA
sin^2a = 3sinA(1 - sin^2A)
sin^2a = 3sinA - 3sin^3A

Step 2:
Rearrange the equation.

sin^3A - 3sinA + sin^2a = 0

Step 3:
Factorize the equation.

sinA(sin^2A - 3) + sin^2a = 0

Step 4:
Since sinA cannot be zero (as it would make the equation undefined), we can divide the equation by sinA.

sin^2A - 3 + (sin^2a / sinA) = 0

Step 5:
Let's substitute tanA = sinA / cosA into the equation.

(tan^2A cos^2A) - 3 + (sin^2a / sinA) = 0
(tan^2A (1 - sin^2A)) - 3 + (sin^2a / sinA) = 0
tan^2A - tan^2A sin^2A - 3 + (sin^2a / sinA) = 0
tan^2A - tan^2A (1 - cos^2A) - 3 + (sin^2a / sinA) = 0
tan^2A - tan^2A + tan^2A cos^2A - 3 + (sin^2a / sinA) = 0
tan^2A cos^2A - 3 + (sin^2a / sinA) = 0

Step 6:
Rearrange the equation.

tan^2A cos^2A + (sin^2a / sinA) - 3 = 0

Step 7:
Let's substitute 1 - cos^2A = sin^2A into the equation.

tan^2A (1 - sin^2A) + (sin^2a / sinA) - 3 = 0
tan^2A - tan^2A sin^2A + (sin^2a / sinA) - 3 = 0
tan^2A - tan^2A + (sin^2a / sinA) - 3 = 0
(sin^2a / sinA) - 3 = 0
sin^2a - 3sinA = 0
sinA (sinA - 3) = 0

Step 8:
To satisfy the equation, either sinA = 0 or sinA - 3 = 0.

Case 1: sinA = 0
If sinA = 0, then cosA = 1 (as sin^2A + cos^2A = 1). So