Prove that (1)/( x - x) - (1)/( x) = (1)/( x) - (1)/( x + x)

Here's the solution for question 30 and 31.

30. Prove that $\frac{1}{\sec x - \tan x} - \frac{1}{\cos x} = \frac{1}{\cos x} - \frac{1}{\sec x + \tan x}$

Step 1: Rearrange the terms to group similar expressions. Move the $\sec x + \tan x$ term to the LHS and the $\cos x$ term to the RHS.

$$\frac{1}{\sec x - \tan x} + \frac{1}{\sec x + \tan x} = \frac{1}{\cos x} + \frac{1}{\cos x}$$

Step 2: Simplify the Right Hand Side (RHS).

$$\text{RHS} = \frac{2}{\cos x}$$

Step 3: Work on the Left Hand Side (LHS). Find a common denominator for the two fractions.

$$\text{LHS} = \frac{(\sec x + \tan x) + (\sec x - \tan x)}{(\sec x - \tan x)(\sec x + \tan x)}$$

Step 4: Simplify the numerator and the denominator. The denominator is a difference of squares.

$$$$