Question

The total number of solutions of $\sin \left\{x\right\}=\cos \left\{x\right\}$ (where $\{.\}$ denotes the fractional part) in $[0,2\pi ]$ is equal to

• A. $5$
• B. $6$
• C. $8$
• D. None of these

Answer 1

The correct option is B $6$

$\sin \left\{x\right\}=\cos \left\{x\right\}$

$\Rightarrow \tan \left\{x\right\}=1$

thus general solution is,

$\left\{x\right\}=x-\left[x\right]=n\pi +\frac{\pi }{4}$, where n is any integer

hence solution in the given interval is,

$x=\frac{\pi }{4},1+\frac{\pi }{4},2+\frac{\pi }{4},3+\frac{\pi }{4},4+\frac{\pi }{4},5+\frac{\pi }{4}$
Hence, option 'B' is correct.