Question

The minimum value of $|\sin x+\cos x+\tan x+\sec x+\text{cosec}x+\cot x|$ is

• A. $2\sqrt{2}-1$
• B. $2\sqrt{2}+1$
• C. $\sqrt{2}-1$
• D. $\sqrt{2}+1$

Answer 1

Answer: (A)

$2\sqrt{2}-1$

Let $sinx+cosx=t$ , we get $t^2-1=sin2x$

Let $P=|sinx+cosx+tanx+cotx+cosecx+secx|=|t+\frac{1+t}{sinxcosx}|=|t+\frac{2}{t-1}|$

$t=\sqrt{2}sin(x+45)$

If $p=1-\sqrt{2}$ , then $P=2\sqrt{2}-1$

Therefore the minimum value is $2\sqrt{2}-1$