If `log_10(sin x) + log_10(tany)+ log_10 2=0` and `coty= 2sqrt3 cos x,` then ordered pair `(x, y)` satisfying the equations simultaneously is(are) (A) `(pi/3 ,pi/3)` (B) `(pi/3 ,pi/6)` (C) `(pi/6 ,(2pi)/3)` (D) `(pi/3 , (7pi)/6)`

Options

A. 0
B. 2
C. 4
D. 8

Answer

C

Explanation

`log_(10)(sin x)+log_(10)(tan y) + log_(10)2=0`
`therefore 2 sin x tan y = 1` ….(1)
where sin x gt 0, tan y gt 0
`cot y=2sqrt(3)cos x` ….(2)
From (1) and (2), `2 sin x = 2sqrt(3)cos x`
`therefore tan x = sqrt(3)`
`rArr x = 2n pi + (pi)/(3), n in I` (as sin x gt 0)
From (1), `tan y =(1)/(sqrt(3)rArr y=n pi + (pi)/(6), n in I`
Hence, ordered pairs are `(pi//3, pi//6), (pi//3,7pi//6),(7pi//3,pi//6), (7pi//3,7pi//6)`.