Question

If $0<x<\frac{\pi }{2}$, then the minimum value of $(\sin x+\cos x+cosec2x)$, is

• A. $27$
• B. $13.5$
• C. $6.25$
• D. None of the above.

Answer 1

The correct option is D None of the above.

we know that $A.M.\ge G.M.$

$⟹\frac{sinx+cosx+cosec2x}{3}\ge \sqrt[3]{3sinx\ast cosx\ast cosec2x}$

$⟹\frac{sinx+cosx+cosec2x}{3}\ge \sqrt[3]{3sinx\ast cosx\ast \frac{1}{sin2x}}$

$⟹sinx+cosx+cosec2x\ge 3\ast \sqrt[3]{3sinx\ast cosx\ast \frac{1}{2sinxcosx}}$

$⟹sinx+cosx+cosec2x\ge 3\ast \sqrt[3]{3\frac{1}{2}}$

$⟹sinx+cosx+cosec2x\ge \frac{3}{2^{\frac{1}{3}}}$

$⟹sinx+cosx+cosec2x\ge 2.3811$