Problem 119 - Algebra and Trigonometry

Question

Verify the identity: $$\frac{1}{\sin x \cos x}-\frac{\cos x}{\sin x}=\tan x$$

Step-by-Step Solution

Answer

The given trigonometric identity \(\frac{1}{\sin x \cos x}-\frac{\cos x}{\sin x}=\tan x\) holds true.

Step 1: Convert to Common Denominator

The first step is to convert the fractions to a common denominator. This allows us to simplify the equation further by combining like terms. The common denominator will be \(\sin x \cos x\). As a result, the expression becomes \(\frac{1}{\sin x \cos x}-\frac{\cos^2 x}{\sin x \cos x}\).

Step 2: Combine Fractions

After getting the common denominator, the next step is to combine the fractions. This results in \(\frac{1 - \cos^2 x}{\sin x \cos x}\).

Step 3: Use Pythagorean Identity

The next step involves using the Pythagorean identity \(\sin^2x + \cos^2x = 1\), which can be rearranged to express \(\cos^2 x\) as \(1 - \sin^2 x\). By substituting into the equation, we get \(\frac{1 - (1 - \sin^2 x)}{\sin x \cos x}\), which simplifies further to \(\frac{\sin^2 x}{\sin x \cos x}\).

Step 4: Simplify Using Trigonometric Identity

Finally, The fraction \(\frac{\sin^2 x}{\sin x \cos x}\) simplifies to \(\frac{\sin x}{\cos x}\), since one of the \(\sin x\) in the numerator cancels out. In trigonometry, \(\frac{\sin x}{\cos x}\) is equivalent to \(\tan x\), which validates the original identity.