Find the general solution of the trigonometric equation `3^(((1)/(2)+log_(3)(cos x+sin x)))-2^(log_(2)(cos x-sin x)=sqrt(2))`

Options

A. `2n pi+(5pi)/(4)`
B. `n pi-(pi)/(4)`
C. `n pi+(-1)^(n)(pi)/(4)`
D. `2n pi+(pi)/(4)`

Answer

A

Explanation

`because A.M. ge G.M. therefore (2^(sin x)+2^(cos x))/(2)ge sqrt(2^(sin x).2^(cos x))`
`therefore 2^(sin x)+2^(cos x)ge 2. sqrt(2^(sin x + cos x))`
But minimum value of cos x + sin x is `- sqrt(2)`
`thereofre 2^(sin x)+2^(cos x)ge 2. sqrt(2^(-sqrt(2)))=2^(1-(1)/(sqrt(2)))`
But the given equation is `2^(sin x)+2^(cos x)=2^(1-(1)/(sqrt(2)))`, which can hold only if `2^(sin x)=2^(cos x)=2^(-(1)/(sqrt(2)))`
`rArr x = 2n pi + (5pi)/(4), n in Z`