If sec x – cos x = 4, then what will be the value of \({{(1 \ + \ cos^2 x)} \over cos x}\)?

Question

Question

This question was previously asked in

SSC CGL 2022 Tier-I Official Paper (Held On : 08 Dec 2022 Shift 3)

Answer 1

Given:

sec x – cos x = 4

Concept Used:

Cos x = \(1\over sec x \)

Calculation:

Let cos x = m

⇒ sec x – cos x = 4

⇒ \(1\over cosx \) - cos x = 4

⇒ \(1\over m \) - m = 4

⇒ 1 - m²= 4m

⇒ m²+ 4m -1 = 0

Finding roots of the above equation:

⇒ m = \( {-b \pm √{b^2-4ac} \over 2a}\)

⇒ m = \( {-4 \pm √{4^2+4} \over 2}\)

⇒ m = \(√5\) - 2

Finding the value:

⇒ \({{(1 \ + \ cos^2 x)} \over cos x}\)

Putting the value of m in above equation:

⇒ \({{(1 \ + \ (√5-2)^2 )} \over√5-2}\) = \({1\over (√5 - 2)} \times {(√5 + 2)\over(√5 + 2) } + √5 - 2 \)

⇒ √5 + 2 + √5 - 2 = \(2{ √{5}}\)

Hence, the value of \(\bf {{(1 \ + \ cos^2 x)} \over cos x}\) is \(2{ √{5}}\) .

Shortcut Trick

We know, if x - 1/x = a then x + 1/x = √(a² + 4)

\({{(1 \ + \ cos^2 x)} \over cos x}\) = secx + cos x = sec x + 1/sec x

Here, similarly sec x - 1/sec x = 4, then sec x + 1/sec x = √(42 + 4) = √20 = 2√5