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Maths-

General

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Question

if cos $left parenthesis x minus y right parenthesis$ , cos x and cos $left parenthesis x plus y right parenthesis$ are in HP, then cos $x times s e c invisible function application fraction numerator y over denominator 2 end fraction$ is equal to

$\sqrt{2}$
$- \sqrt{2}$
$\pm \sqrt{2}$
$None of these$

The correct answer is: $plus-or-minus square root of 2$

Given cos $left parenthesis x minus y right parenthesis$ , cos x and cos $left parenthesis x plus y right parenthesis$ are in HP. So
$fraction numerator 2 over denominator cos invisible function application x end fraction equals fraction numerator 1 over denominator c o s invisible function application left parenthesis x minus y right parenthesis end fraction plus fraction numerator 1 over denominator c o s invisible function application left parenthesis x plus y right parenthesis end fraction$
$rightwards double arrow fraction numerator 2 over denominator cos invisible function application x end fraction equals fraction numerator 2 c o s invisible function application x times c o s invisible function application y over denominator c o s to the power of 2 end exponent invisible function application x minus s i n to the power of 2 end exponent invisible function application y end fraction$
$rightwards double arrow c o s to the power of 2 end exponent invisible function application x minus s i n to the power of 2 end exponent invisible function application y equals c o s to the power of 2 end exponent invisible function application x times c o s invisible function application y$
$rightwards double arrow c o s to the power of 2 end exponent invisible function application x times 2 s i n to the power of 2 end exponent invisible function application fraction numerator y over denominator 2 end fraction equals 4 s i n to the power of 2 end exponent invisible function application fraction numerator y over denominator 2 end fraction times c o s to the power of 2 end exponent invisible function application fraction numerator y over denominator 2 end fraction$
$rightwards double arrow c o s invisible function application x times s e c invisible function application fraction numerator y over denominator 2 end fraction equals plus-or-minus square root of 2$

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