Question

Proving Identities

Verify the identity$\frac{2\sin x\cos x}{{(\sin x+\cos x)}^2-1}=1$

Short answer

The expression$\frac{2\sin x\cos x}{{(\sin x+\cos x)}^2-1}=1$ is an identity.

Step-by-step solution

Step 1- Given information.

An expression$\frac{2\sin x\cos x}{{(\sin x+\cos x)}^2-1}=1$

Step 2. Concept.

In this problem, separate the expression into two parts: LHS and RHS. After that, simplify them one by one. The phrase is an identity if both responses are equal.

Step 3. Calculation.

Now, simplify LHS of $\frac{2\sin x\cos x}{{(\sin x+\cos x)}^2-1}=1$:

We use identities such as-

$$\begin{gathered} {(a+b)}^2=a^2+b^2+2ab \\ =\frac{2\sin x\cos x}{{(\sin x+\cos x)}^2-1} \\ =\frac{2\sin x\cos x}{{\sin }^{2}x+\cos x^2+2\sin x\cos x-1} \\ =\frac{2\sin x\cos x}{1+2\sin x\cos x-1} \\ =\frac{2\sin x\cos x}{2\sin x\cos x} \\ =1 \end{gathered}$$

RHS of the equation is 1

Both RHS and LHS are equal. Hence it is an identity