Can you get the hint?

Well, it is kind of hard to think about this. The fastest way out of this problem is: $\large 21cot^{2}x = -25$ $\large (25 - 4)cot^{2}x = -25$ $\large 25cot^{2}x + 25 = 4cot^{2}x$ $\large cot^{2}x + 1 = \frac{4cot^{2}x}{25}$ $\large cossec^{2}x = \frac{4cot^{2}x}{25}$ $\large cossecx = \frac{2cotx}{5}$ $\large sinx = \frac{5}{2cotx}$ $\large sinx = \frac{5tanx}{2}$ $\large \frac{sinx}{tanx} = \frac{5}{2}$ $\large \boxed{cosx = 2.5}$ Considering that the max "cos" or "sin" of any real number is 1, we can assume that x is not a real number.

Since $\cot^2 x \geq 0$ and $21>0$ , L.H.S is positive and R.H.S is negative and this is possible only if $x$ is a complex number.

Observe that $21$ is positive and that $\cot^2 x$ is positive when $x \in R$ . Hence, the LHS is positive. But the RHS is negative, indicating that our assumption that $x \in R$ is false.

Substituting the trigonometric functions of x, one can solve this problem, but it is a longer way:

$21cot^{2}x = -25 \Rightarrow 21 (\frac{cosx}{sinx})^{2} = -25 \Rightarrow \frac{21cos^{2}x}{sin^{2}x} = -25$ $\Rightarrow 21cos^{2}x = -25sin^{2}x \Rightarrow cosx = sinx \sqrt{\frac{-25}{21}}$ $\Rightarrow cosx = \frac{sinx \times 5i \sqrt{21}}{21}$ . We can already note that x is not a real number, but let's continue:

$cot^{2}x = \frac{-25}{21}$ . Using the trigonometric identity $1 + cot^{2}x = csc^{2}x$ , we get that $1 + (\frac{-25}{21}) = csc^{2}x \Rightarrow csc^{2}x = \frac{-4}{21}$ $\Rightarrow sinx = \frac{\sqrt{21}}{2i}$ $\therefore cosx = \frac{\frac{\sqrt{21}5i \sqrt{21}}{2i}}{21}$ $\Rightarrow cosx = \frac{\frac{21 \times 5}{2}}{21} \Rightarrow \boxed{cosx = 2.5}$

Note that the value of $cosx$ is greater than the maximum possible value of a function of the form $f(x) = cosx$ , so it is clear that x is not a real number, but I still haven't got enough of it:

$cos^{2}x + sin^{2}x = 1 \Rightarrow (\frac{5}{2})^2 + sin^{2}x = 1 \Rightarrow sin^{2}x = \frac{10-25}{10} \Rightarrow sinx = \sqrt{\frac{-15}{10}} \Rightarrow \boxed{sinx = i \sqrt{\frac{15}{10}}}$

We are now sure that x is not a real number.