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Chapter 5: Problem 4

Verify each identity. $$\cot (-x) \sin x=-\cos x$$

Short Answer

Expert verified
Therefore, the identity \(\cot (-x) \sin x=-\cos x\) has been successfully verified.

Step by step solution

01

Substitute the cotangent function

We begin by expressing the cotangent function in terms of sine and cosine. The cotangent of an angle is the cosine of that angle divided by the sine of that angle, but since it's \(-x\) we will have to reflect the function over the y-axis. So, \(\cot(-x) = \frac{\cos(-x)}{\sin(-x)}\).
02

Replace \(-x\) with \(x\)

When it comes to cosine, it is an even function, meaning that \(\cos(-x) = \cos(x)\). However, sine is an odd function, which implies \(\sin(-x) = -\sin(x)\). So, now our equation looks like this: \(\frac{\cos(x)}{-\sin(x)} \cdot \sin(x) = -\cos(x)\).
03

Simplify the Equation

Now we simplify the equation by multiplying both sides of the equation. On the left side, \(\sin(x)\) in the demininator and numerator cancels out and we are left with \( -\cos(x) \). So, \( -\cos(x) = -\cos(x) \).

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