Solving sin(x)^sin(x)=2

We have two exponential equations with trigonometric functions (sin(x))^(sin(x))=2 and (sin(x))^(cos(x))=2. The tetration equation (sin(x))^sin(x)=2 requires us to use the Lambert W function, complex exponential, complex logarithm, and the quadratic formula. However, for (cos(x))^cos(x)=2, we can argue it has real solutions by intermediate value theorem but I do not know a way to solve it algebraically. I know we can use Newton's method to get an approximation for the solution but I wonder if we can somehow also use the Lambert W function to find a closed-form of the solution.

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0:00 I have a math conundrum
0:12 solving (sin(x))^sin(x)=2
7:40 why (sin(x))^cos(x)=2 has real solutions
10:16 can WolframAlpha solve (sin(x))^cos(x)=2?

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