For $0\lt\theta\lt\frac{\pi}{4}$, let $a=\left((\sin \theta)^{\sin \theta}\right)\left(\log _{2} \cos \theta\right), b=\left((\cos \theta)^{\sin \theta}\right)\left(\log _{2} \sin \theta\right), c=\left((\sin \theta)^{\cos \theta}\right)\left(\log _{2} \cos \theta\right)$ and $d=\left((\sin \theta)^{\sin \theta}\right)\left(\log _{2} \sin \theta\right)$. Then, the median value in the sequence $a, b, c, d$ is

In the graph given below, the green line shows the sine curve and the blue line shows the cosine curve.

By closely looking at the graph, we can conclude that for $0< \theta < \dfrac{\pi}{4}$, the value of $\cos \theta$ will be higher than the value of $\sin \theta$.

To make our calculations slightly easier, let us assume some values for $\sin \theta$ and $\cos \theta$.

Let: $\sin \theta = 0.01$ and $\cos \theta = 0.1$.

$\Rightarrow \dfrac{a}{b} = \dfrac{(0.01)^{0.01}}{(0.1)^{0.01}} \times \dfrac{\log_2 0.1}{\log_2 0.01}$

$\Rightarrow (\dfrac{0.01}{0.1})^{0.01} \times \log_{0.01} 0.1$ [$\because \dfrac{\log a}{\log b} = \log_b a$]

$\Rightarrow (\dfrac{1}{10})^{0.01} \times (\dfrac{-1}{-2}) \log_{10} 10$ [$\because \log_{m^b} n^a = \dfrac{a}{b} \log_m n$]

$\Rightarrow \dfrac{a}{b} = (\dfrac{1}{10})^{0.01} \times \dfrac{1}{2}(1)$ [$\because \log_a a = 1$]

or,

$\dfrac{a}{b} = \dfrac{1}{10^{0.01}} \times \dfrac{1}{2}$

$\therefore b > a$

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Using the same approach, we'll compare $c$ and $d$.

$\Rightarrow \dfrac{c}{d} = \dfrac{(0.01)^{0.1}}{(0.01)^{0.01}} \times \dfrac{1}{2}$

$\Rightarrow \dfrac{c}{d} = (0.01)^{0.1-0.01} \times \dfrac{1}{2}$

$\Rightarrow \dfrac{c}{d} = (\dfrac{1}{100})^{0.09} \times \dfrac{1}{2}$

$\therefore d > c$

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Similarly, comparing $b$ and $d$, we get $b > d$.

$\because b > a, d > c$ and $b > d$, it can be said that $b$ is the greatest among all.

Now comparing $a$ and $c$, we get $a > c$. Thus, $c$ is the smallest value among all.

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The 'median' of a sequence is the middle-most value, given that the sequence is arranged in an ascending order.

$b$ (the greatest value) and $c$ (the smallest value) will occupy the extreme positions when arranged in an increasing order. In other words, they cannot occupy the positions in the middle.

Hence, $a$ and $d$ will occupy the middle positions.