Solve the given equation (tan x)/(sec x-1)+(sin x)/(1-cos x)=2cosec x

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Solve the given equation  (tan x)/(sec x-1)+(sin x)/(1-cos x)=2cosec x

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Simplify left-hand side: We will start by simplifying the left-hand side of the equation. The first term (tan x)/(sec x-1) can be rewritten using the identity tan x = sin x / cos x and sec x = 1 / cos x. This gives us (sin x / cos x) / (1/cos x - 1).Combine terms: Simplify the expression further by multiplying the numerator and denominator by cos x to get rid of the fraction in the denominator. This results in (sin x / cos x) * (cos x / (1 - cos x)), which simplifies to sin x / (1 - cos x).Simplify right-hand side: Now, let's look at the second term (sin x)/(1-cos x). We notice that it is identical to the simplified form of the first term. Therefore, we can combine them to get 2 * (sin x / (1 - cos x)).Verify identity: Next, we will simplify the right-hand side of the equation. The identity cosec x = 1 / sin x allows us to rewrite 2cosec x as 2 / sin x.Rewrite expressions: Now we have 2 * (sin x / (1 - cos x)) on the left-hand side and 2 / sin x on the right-hand side. To verify the identity, we need to show that these two expressions are equal.Common denominator: We can rewrite the left-hand side as 2sin x / (1 - cos x) and the right-hand side as 2sin x / sin^2 x. To compare these two expressions, we need to have a common denominator.Apply identity: We know that sin^2 x + cos^2 x = 1, which means sin^2 x = 1 - cos^2 x. We can use this identity to rewrite the denominator of the right-hand side expression as 1 - cos^2 x.Simplify further: Now the right-hand side becomes 2sin x / (1 - cos^2 x). Since 1 - cos^2 x is the same as sin^2 x, we can simplify this to 2sin x / sin^2 x, which simplifies further to 2 / sin x.Confirm equality: We see that both sides of the equation have the same expression 2 / sin x, which means the identity (tan x)/(sec x-1)+(sin x)/(1-cos x)=2cosec x holds true.

Solve the given equation $\frac{\tan x}{\sec x-1}+\frac{\sin x}{1-\cos x}=2 \operatorname{cosec} x$

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Solve the given equation $\frac{\tan x}{\sec x-1}+\frac{\sin x}{1-\cos x}=2 \operatorname{cosec} x$