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If cos ( x - y ) , cos x and cos ( x + y ) are in HP , then cos x sec ( y / 2 ) is equal to

Options

  1. A± 2
  2. B± 1 / 2
  3. C± 2
  4. DNone of these

Correct answer

A. ± 2

Step-by-step solution

Given that, cos ( x - y ) , cos x , cos ( x + y ) are in HP . Then, cos x = 2 cos ( x - y ) cos ( x + y ) cos ( x - y ) + cos ( x + y ) ⇒ cos x = 2 cos 2 x - sin 2 y 2 cos x cos y ⇒ cos 2 x cos y = cos 2 x - sin 2 y ⇒ cos 2 x ( cos y - 1 ) = - sin 2 y ⇒ cos 2 x ( 1 - cos y ) = 1 - cos 2 y ⇒ cos 2 x ( 1 - cos y ) = ( 1 - cos y ) ( 1 + cos y ) ⇒ cos 2 x = 1 + cos y ⇒ cos 2 x = 2 cos 2 y / 2 ⇒ cos 2 x sec 2 ( y / 2 ) = 2 ⇒ cos x sec ( y / 2 ) = ± 2

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