QJEEM17Ma656c: JEE Main 2017 Mathematics - Class 11 JEEM17Ma656c Solution: If , then the value of is :

Question

Verified Solution • 2025–26 Session moderate

If

5\left( {{{\tan }^2}x - {{\cos }^2}x} \right) = 2\cos 2x + 9

,

then the value of

\cos 4x

is :

Accepted Answer

Given that,

5\left( {{{\tan }^2}x - {{\cos }^2}x} \right) = 2\cos 2x + 9

\Rightarrow 5\left( {{{{{\sin }^2}x} \over {{{\cos }^2}x}} - {{\cos }^2}x} \right) = 2\left( {2{{\cos }^2}x - 1} \right) + 9

Let

{\cos ^2}x = t,

then we have

5\left( {{{1 - t} \over t} - t} \right) = 2\left( {2t - 1} \right) + 9

\Rightarrow 5\left( {{{1 - t - {t^2}} \over t}} \right) = 4t - 2 + 9

\Rightarrow 5 - 5t - 5{t^2} = 4{t^2} + 7t

\Rightarrow 9{t^2} + 12t - 5 = 0

\Rightarrow 9{t^2} + 15t - 3t - 5 = 0

\Rightarrow 3t\left( {3t + 5} \right) - 1\left( {3t + 5} \right) = 0

\Rightarrow \left( {3t + 5} \right)\left( {3t - 1} \right) = 0

\therefore

t = {1 \over 3}

and

t = - {5 \over 3}

If

t = - {5 \over 3}

then

{\cos ^2}x

is negative

So ,

t

can not be

- {5 \over 3}

.

So, correct value of

t = {1 \over 3}

then

\cos {}^2x = t = {1 \over 3}

\therefore\,\,\,

\cos 4x

= 2{\cos ^2}2x - 1

= 2{\left[ {2{{\cos }^2}x - 1} \right]^2} - 1

= 2.{\left[ {2.{1 \over 3} - 1} \right]^2} - 1

= 2.{\left( { - {1 \over 3}} \right)^2} - 1

= {2 \over 9} - 1

= - {7 \over 9}

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