Question 880271
I'm not sure there is an analytical solution. I spent a few minutes trying to derive one but gave up and solved it by iteration. That is, select an initial value of theta and iterively converge to a more precise value.
Using x instead of theta, the given expression is
(1) {{{sin^4(x) = cos^5(x))}}} or by dividing each side by {{{cos^4(x)}}} we get
(2){{{tan^4(x) = cos(x)}}}
Since the absolute value of the cosine function is less than or equal to one, and the tan(45) is one, start x at 45 degrees and get
(3) at x = 45, LS = 1, RS = .335, note that LS is greater than the RS   
Now decrease x, 
(4) at x = 40, LS = .4957, RS = .766, note LS is now less than RS, we went too far, so x lies between 40 and 45.  
(5) at x = 43, LS = .756, RS = .731, LS is greater than RS, now x is between 40 and 43.      
(6) at x = 42.5 LS = .7050, RS = .7372 LS is less than RS, went too far, now x is between  42.5 and 43.    
Keep doing this until you get to  
(7) x = 42.78634   
To check this use (1) 
Is ({{{sin^4(42.78634) = cos^5(42.78634))}}})?  
Is (0.212892 = 0.212892)? Yes
Answer; theta equals approximately 42.78634 degrees.