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Let x, y, in R satisfy the condition suc...

Let `x, y, in R` satisfy the condition such that sin x sin y + 3 cos y +4 sin y cos `x=sqrt(26)`. The value of `tan^(2)x + cot^(2)y` is equal to

A

`9xx17`

B

205

C

`(1)/(16)+(9)/(17)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
`sin x sin y + 3 cos y + 4 sin y cos x = sqrt(26)`
`therefore 3 cos y + (sin x + 4 cos x) sin y = sqrt(26)`
`therefore 3 cos y + (sin x + 4 cos x)sin y`
`le sqrt(9+(sin x+4 cos x)^(2))`
`le sqrt(9+1+16)`
`= sqrt(26)`
`therefore sin x sin y + 3 cos y + 4 sin y cos x = sqrt(26)`
`rArr sin x sin y = (cos y)/(3) = (sin y cos x)/(4)`
`rArr 3 tna y = cosec x` and `tan x = 1//4`
`rArr 9 tan^(2)y = cosec^(2)x=(1+cot^(2)x)=17`
`rArr tan^(2) x + cot^(2)y=(1)/(16)+(9)/(17)`
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