Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity. $$\frac{\csc (-x)}{\sec (-x)}=-\cot x$$

Step-by-Step Solution

Answer

Given identity \(\frac{\csc (-x)}{\sec (-x)}=-\cot x\) is true as both the sides of the equation have been proven to be equal.

Step 1: Express in terms of sine, cosine and cotangent

Rewrite \(\csc (-x)\) and \(\sec (-x)\) in terms of sine and cosine. By definition, \(\csc (-x) = \frac{1}{\sin(-x)}\) and \(\sec(-x) = \frac{1}{\cos(-x)}\). Thus the given identity becomes: \(\frac{1}{\sin(-x)}*\frac{\cos(-x)}{1}= -\cot x\)

Step 2: Simplify Expression

The expression can be simplified to \(\frac{\cos(-x)}{\sin(-x)} = -\cot x\). Now we multiply both the sides by -1 to remove the negatives, the equation transforms to: \(-\frac{\cos(-x)}{\sin(-x)} = \cot x\)

Step 3: Using Negative Angle Identities

Using the negative angle identities, which state that \(\cos(-x) = \cos x\) and \(\sin(-x) = -\sin x\), we can rewrite the left side as \(-\frac{\cos(x)}{-\sin(x)}\). This simplifies to \(\frac{\cos(x)}{\sin(x)}\).

Step 4: Finalize the Transformation

The expression \(\frac{\cos(x)}{\sin(x)}\) is equivalent to \(\cot x\), which matches the right side of the original equation.