Question

Question: Proving Identities

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

Short answer

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

Step-by-step solution

Step 1. Given information.

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

Step 2. Concept used.

Do two time simplification. First time, simplify the LHS part of the expression and second time simplify the RHS part of the expression. If both come out be equal then the expression is an identity.

Step 3. Calculation.

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

$$\begin{gathered} {(\sin x+\cos x)}^2{(\sin x+\cos x)}^2={\sin }^{2}x+{\cos }^{2}x+2\sin x\cos x \\ \because {\sin }^{2}x+{\cos }^{2}x=1 \\ {(\sin x+\cos x)}^2=1+2\sin x\cos x \end{gathered}$$

RHS of the equation is$1+2\sin x\cos x$

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