Question

Prove that : \(\frac{1}{\sec x - \tan x} - \frac{1}{\cos x} = \frac{1}{\cos x} - \frac{1}{\sec x + \tan x}\)

Hint:

Rearranging terms is a powerful algebraic trick to simplify trigonometric proofs. It allows you to use conjugate identities like \((\sec x - \tan x)(\sec x + \tan x) = 1\) directly in a single step!

Answer 1

Step 1: Understanding the Question:
The topic of this question is Trigonometric Identities.
We are asked to prove a standard trigonometric equality containing reciprocal and fractional terms.
Instead of evaluating the complicated Left-Hand Side (LHS) and Right-Hand Side (RHS) individually, we can rearrange the terms to simplify the proof structure.

Step 2: Key Formula or Approach:
We can rearrange the equation by grouping the secant and tangent fractions on one side and the cosine fractions on the other:
\[ \frac{1}{\sec x - \tan x} + \frac{1}{\sec x + \tan x} = \frac{1}{\cos x} + \frac{1}{\cos x} \] We will prove this equivalent statement, which simplifies to:
\[ \frac{1}{\sec x - \tan x} + \frac{1}{\sec x + \tan x} = \frac{2}{\cos x} = 2 \sec x \] We will use the Pythagorean identity:
\[ \sec^2 x - \tan^2 x = 1 \]

Step 3: Detailed Explanation:

• Consider the Left-Hand Side (LHS) of the rearranged equation:
\[ \text{LHS} = \frac{1}{\sec x - \tan x} + \frac{1}{\sec x + \tan x} \]

• Take a common denominator to add the two fractions:
\[ \text{LHS} = \frac{(\sec x + \tan x) + (\sec x - \tan x)}{(\sec x - \tan x)(\sec x + \tan x)} \]

• Simplify the numerator:
\[ (\sec x + \tan x) + (\sec x - \tan x) = 2 \sec x \]

• Simplify the denominator using the difference of squares:
\[ (\sec x - \tan x)(\sec x + \tan x) = \sec^2 x - \tan^2 x \]

• Substitute the Pythagorean identity \(\sec^2 x - \tan^2 x = 1\):
\[ \text{LHS} = \frac{2 \sec x}{1} = 2 \sec x \]

• Simplify the Right-Hand Side (RHS) of the rearranged equation:
\[ \text{RHS} = \frac{1}{\cos x} + \frac{1}{\cos x} = \frac{2}{\cos x} = 2 \sec x \] Since the rearranged LHS is equal to the rearranged RHS, the original identity is proven.


Step 4: Final Answer:
Hence, the identity is proved.