驗證 cot x · sin x = cos x

Problem

Verify the trigonometric identity

$$\cot x \cdot \sin x = \cos x.$$

Step 1: Understand the Goal

We need to show that the left-hand side of the equation is equal to the right-hand side.

The identity is:

$$\cot x \cdot \sin x = \cos x.$$

To verify it, we start with the left-hand side:

$$\cot x \cdot \sin x.$$

Step 2: Use the Quotient Identity

We use the quotient identity for cotangent:

$$\cot x = \frac{\cos x}{\sin x}.$$

Substitute this into the left-hand side:

$$\cot x \cdot \sin x = \frac{\cos x}{\sin x} \cdot \sin x.$$

Step 3: Cancel the Common Factor

Now simplify:

$$\frac{\cos x}{\sin x} \cdot \sin x.$$

The factor $\sin x$ appears in the denominator and is also being multiplied, so it cancels:

$$\frac{\cos x}{\sin x} \cdot \sin x = \cos x.$$

Step 4: Final Answer

After simplifying the left-hand side, we get:

$$\cos x.$$

This matches the right-hand side of the original identity:

$$\cot x \cdot \sin x = \cos x.$$

Therefore, the identity is verified.