$\frac{\sin^{4} x + \cos^{4} x}{\sin x \cos x}$

= $\frac{(\sin^{2}x + \cos^{2}x)^{2} - 2 \sin^{2} x \cos^{2} x}{\sin x \cos x}$

= $\frac{2-4\sin^{2} x \cos^{2} x}{2 \sin x \cos x}[ as (\sin^{2}x + \cos^{2}x =1) ]$

= $\frac{2-(2 \sin x \cos x)^{2}}{2 \sin x \cos x}$

= $\frac{2-\sin^{2}2x}{\sin 2x}$

Now the value of the function to be # minimum the Denominator of the function must be # Maximum.

$Max(\sin 2x) = 1 [as (0<=\sin \theta <=1 )for all (1<\theta<\frac{\pi}{2})]$

$Min (\frac{\sin^{3} x}{\cos x}+\frac{\cos^{3}}{\sin x})$

= $\frac{2-\sin^{2}2x}{\sin 2x}$

= $\frac{2 - 1}{1} (putting , \sin 2x =1 )$

= $1$

The sequences $(\sin^{3} x,cos^{3} x)$ and $(\frac{1}{\cos x},\frac{1}{\sin x})$ are ordered oppositely. So, by using the Chebyshev's Sum Inequality on these by creating a scalar product matrix, we can see that the minimum value is $\sin^{2} x+\cos^{2} x$ which is equal to $\huge{1}$ .

As $sinx>=0$ and $cosx>=0$ for the interval [ $0$ , $\frac{\pi}{2}$ ], using the inequality of arithmetic and geometric means: $\frac{\frac{sin^{3}x}{cosx}+\frac{cos^{3}x}{sinx}}{2}>=\sqrt{\frac{sin^{3}x}{cosx}\times\frac{cos^{3}x}{sinx}}=sinxcosx$ then $\frac{sin^{3}x}{cosx}+\frac{cos^{3}x}{sinx}>=sin(2x)$ As the minimum for that inequality ocurs when all the terms are equal, then we have $sinx=cosx$ and $x=\frac{\pi}{4}$ so $sin(2x)$ will be $1$ in that case and the minimum value for the expression will be $1$ .

solve it it comes out to be cotx as simple as that ans is 1

By AM-GM inequality, this is larger than or equal to 2sinxcosx. Equality is obtained when sinx=cosx. This value is 2sin(pi/4)cos(pi/4)=1

If $x=a$ is an extrema then $x=\frac{\pi}{2}-a$ is as well. Looking at the image, the graph has a single minimum. Therefore, the only possible value for $a$ is one such that $a = \frac{\pi}{2}-a$ , so $a = \frac{\pi}{4}$ .

At the point $x = a$ , $\sin{x}=\cos{x}$ so the expression becomes:

$\sin^2{x} + \cos^2{x}=\boxed{1}$

when i dunno how to slove the quiz,i put 1....

we can intuitively solve this problem by looking at the graph of this question.

arithmatic mean is always greater than or equal to the geometric mean of given set of positive numbers. We can use this property to solve the question

Instinct and Probability are the two factors which provides me correct answers to DIFFICULT problems. Actually, I would just like to share it with you guys. According to my experience and observation, numbers 0,1, 2, 5, 10 and 100 in the choices are 50% certain to be correct answers in this tricky questions. It is in the strength of your instinct to go with that answer... Dont forget to follow me if I am correct because I might teach how to do probability..... hahaha! See ya!

substitute any angle between o and 90 in place of x in the expression

We solve the equation and get 1/sinxcosx - 2sinxcosx which is 2/sin2x -sin2x ie 2cosec2x-sin2x. To get minimum value, we must consider greatest value of neg term and smallest of positive term. So, we get 2(1)-1=1 as the smallest value