Problem 36 - Precalculus Essentials

Question

Verify each identity. $$\frac{\csc x-\sec x}{\csc x+\sec x}=\frac{\cot x-1}{\cot x+1}$$

Step-by-Step Solution

Answer

After simplifying the equation using trigonometric identities, it's evident that \(\frac{\csc x-\sec x}{\csc x+\sec x}=\frac{\cot x-1}{\cot x+1}\). Thus, the identity is verified.

Step 1: Rewrite Using Pythagorean Identities

Start by replacing \(\csc x\) and \(\sec x\) with their reciprocal identities. Here \(\csc x = \frac{1}{\sin x}\) and \(\sec x = \frac{1}{\cos x}\). Thus, the equation becomes \(\frac{\frac{1}{\sin x}-\frac{1}{\cos x}}{\frac{1}{\sin x}+\frac{1}{\cos x}}\).

Step 2: Simplify the Equation

Next, the equation can be rewritten with a common denominator. This gives, \(\frac{\frac{\cos x-\sin x}{sin x \cos x}}{\frac{\cos x+\sin x}{sin x \cos x}}\). The common factors on the numerator and denominator will cancel out and we get \(\frac{\cos x-\sin x}{\cos x+\sin x}\).

Step 3: Rewrite Using Reciprocal Identities

Now replace \(\cos x\) as \(1/\sec x\) and \(\sin x\) as \(1/\csc x\). This gives \(\frac{1/\cot x -1}{1/\cot x +1}\).

Step 4: Simplify Further

Simplify the expression to get it in terms of \(\cot x\). This results in the expression: \(\frac{\cot x - 1}{\cot x + 1}\).