sinA+2cosA/sin2A+2sinA*cosA=cosecA

Solving the Trigonometric Equation sinA+2cosA/sin2A+2sinA*cosA=cosecATo solve the trigonometric equation sinA+2cosA/sin2A+2sinA*cosA=cosecA, we can start by simplifying the left-hand side of the equation using trigonometric identities and then verifying if it equals the right-hand side.First, let’s simplify the left-hand side of the equation:sinA + 2cosA / (sin2A + 2sinA*cosA)We can use the following trigonometric identities to simplify this expression:sin2A = 2sinA*cosAcosecA = 1/sinAUsing these identities, we can rewrite the left-hand side of the equation as follows:sinA + 2cosA / (2sinAcosA + 2sinAcosA)This simplifies to:sinA + 2cosA / 4sinA*cosANow, let’s simplify further by expressing sin and cos in terms of tan:sinA = tanA / √(1 + tan^2 A) cosA = 1 / √(1 + tan^2 A)Substituting these expressions into our equation, we get:tanA/√(1+tan^2 A) + 2/√(1+tan^2 A) / 4tanA/√(1+tan^2 A)*1/√(1+tan^2 A)This simplifies to:(tanA + 2) / (4tan^2 A)Now, we can express cosec A in terms of sin A:cosec A = 1/sin ASubstituting this into our equation, we get:1/(tan A) = (tan A + 2) / (4tan^2 A)Cross multiplying gives us:tan^3 A = tan^3 A + 2tan^2 ASubtracting tan^3 A from both sides yields:0 = 2tan^2 ADividing both sides by 2 gives us:tan^2 A = 0Taking the square root of both sides gives us:tan A = 0Therefore, the solution to the given trigonometric equation is tan A = 0. Sorry if it’s complicated ;-;