Question

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

Hint:

In proofs like this, if standard simplification is messy, try shifting terms to make the equation symmetric. Proving \(A - B = C - D\) is the same as proving \(A + D = B + C\).

Answer 1

Step 1: Start with LHS
LHS = 1/(sec x − tan x) + 1/(sec x + tan x)

Step 2: Take LCM
LHS = [(sec x + tan x) + (sec x − tan x)] / [(sec x − tan x)(sec x + tan x)]

Simplify numerator:
= (sec x + tan x + sec x − tan x)
= 2 sec x

Simplify denominator using identity:
(sec x − tan x)(sec x + tan x)
= sec²x − tan²x
= 1

So,
LHS = 2 sec x
= 2 / cos x

Step 3: Evaluate RHS
RHS = 1/cos x + 1/cos x
= 2/cos x

Step 4: Conclusion
LHS = RHS

Final Answer:
Hence proved.