If tan (π cos θ) = cot (π sin θ), $0<\theta<\frac{\pi}{2}$; then what is the value of $8 \sin ^2\left(\theta+\frac{\pi}{4}\right)$ ?

If $\tan \alpha=\frac{1}{7}$, $\sin \beta=\frac{1}{\sqrt{10}}$; $0<\alpha, \beta<\frac{\pi}{2}$, then what is the value of cos (α + 2β) ?

Consider the following for the next items that follow:

Consider the equation (1 - x)4 + (5 - x)⁴ = 82.

What is the number of real roots of the equation?

Consider the following for the next items that follow:

Consider the equation (1 - x)4 + (5 - x)⁴ = 82.

What is the sum of all the roots of the equation?