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If 5 (tan ^(2) x - cos ^(2) x ) = 2 cos ...

If `5 (tan ^(2) x - cos ^(2) x ) = 2 cos 2x +9`, then the value of `cos 4 x ` is

A

`-(3)/(4)`

B

`(1)/(3)`

C

`(2)/(9)`

D

`-(7)/(9)`

Text Solution

Verified by Experts

The correct Answer is:
D
Given , `5 (tan ^(2) x - cos^(2) x) 2 cos 2x + 9`
`rArr 5((2sin^(2)x)/(2cos2x) -cos^(2) x) = 2cos 2x + 9`
` rArr 5((1-cos2x)/(1+ cos 2x)-(1+ cos2x)/(2))= 2 cos 2x + 9`
Put `cos 2x = y,` we have
`5((1-y)/(1+y)-(1+cos2x)/(2)) = 2y + 9`
`rArr 5(2-2y -1-y^(2) -2y)=2(1+y)(2y+9)`
` rArr 5(1-4y -y^(2))=2(2y + 9+ 2y^(2) + 9y)`
`9y^(2) + 42 + 13 = 0`
`rArr 9y^(2) + 3y + 39y + 13=0`
`3y(3y + 1)+(13(3y + 1)=0`
`rArr (3y + 1)(3y + 13) =0`
`rArr y = -(1)/(3),-(13)/(3)`
`therefore cos2x =- (1)/(3),-(13)/(3)`
`therefore cos 2x = - (1)/(3)" "[therefore cos 2x ne -(13)/(3)]`
Now, `cos4x = 2 cos^(2)2x-1`
`= 2(-(1)/(3))^(2) -1 = (2)/(9) -1=-(7)/(9)`
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