Question

Proving Identities

Verify the identity$\cos (-x)-\sin (-x)=\cos x+\sin x$

Short answer

The expression$\cos (-x)-\sin (-x)=\cos x+\sin x$ is an identity.

Step-by-step solution

Step 1. Given information.

An expression as$\cos (-x)-\sin (-x)=\cos x+\sin x$

Step 2. Concept used.

Divide the phrase into two parts: LHS and RHS. solve or simplify each one independently. If the results are the same, the supplied phrase is an identity.

Step 3. Calculation.

Now, simplify LHS of $\cos (-x)-\sin (-x)=\cos x+\sin x$:

$\cos (-x)-\sin (-x)$

Here we will use some trigonometric properties for verifying the equation.

The properties are-

$$\begin{gathered} \cos (-x)=\cos x \\ \& \\ \sin (-x)=-\sin x \end{gathered}$$

Now further calculation-

$$\begin{gathered} \cos (-x)-\sin (-x) \\ =\cos x-(-\sin x) \\ =\cos x+\sin x \end{gathered}$$

RHS of the equation is-

$\cos x+\sin x$

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