The graph of the LHS gets nowhere near the value on the RHS. It looks as if all the solutions are complex:
>> syms x
>> s=solve(sin(x)^10+cos(x)^10 ==61/256)
s =
-log(-(- (3*13^(1/2))/2 - 11/2)^(1/4))*1i
-log(-((3*13^(1/2))/2 - 11/2)^(1/4))*1i
-log(-(- (3*13^(1/2))/2 - 11/2)^(1/4)*1i)*1i
-log((- (3*13^(1/2))/2 - 11/2)^(1/4)*1i)*1i
-log(-((3*13^(1/2))/2 - 11/2)^(1/4)*1i)*1i
-log(((3*13^(1/2))/2 - 11/2)^(1/4)*1i)*1i
-log(- 3^(1/2)/2 - 1i/2)*1i
-log(- 3^(1/2)/2 + 1i/2)*1i
-log(3^(1/2)/2 - 1i/2)*1i
-log(3^(1/2)/2 + 1i/2)*1i
-log(- (3^(1/2)*1i)/2 - 1/2)*1i
-log(1/2 - (3^(1/2)*1i)/2)*1i
-log((3^(1/2)*1i)/2 - 1/2)*1i
-log((3^(1/2)*1i)/2 + 1/2)*1i
-log((- (3*13^(1/2))/2 - 11/2)^(1/4))*1i
-log(((3*13^(1/2))/2 - 11/2)^(1/4))*1i
>> eval(s)
ans =
-2.35619449019234 - 0.597381608643555i
-2.35619449019234 + 0.597381608643553i
-0.785398163397448 - 0.597381608643555i
2.35619449019234 - 0.597381608643555i
-0.785398163397448 + 0.597381608643553i
2.35619449019234 + 0.597381608643553i
-2.61799387799149 + 5.55111512312578e-17i
2.61799387799149 + 5.55111512312578e-17i
-0.523598775598299 + 5.55111512312578e-17i
0.523598775598299 + 5.55111512312578e-17i
-2.0943951023932 + 5.55111512312578e-17i
-1.0471975511966 + 5.55111512312578e-17i
2.0943951023932 + 5.55111512312578e-17i
1.0471975511966 + 5.55111512312578e-17i
0.785398163397448 - 0.597381608643555i
0.785398163397448 + 0.597381608643553i
The results from Wolfram alpha are also all complex, but include integral multiples of portions, so I think Matlab is showing only some of the infinite number of solutions, as it is indeed a cyclic function.
what I find
