Explanation

8sin(theta)-4=cos(theta); squaring gives 65sin^2(theta)-64sin(theta)+15=0, so sin(theta)=3/5 or 5/13. Only sin(theta)=3/5 (cos(theta)=4/5) satisfies the unsquared equation (the other gives 28/13, not 4). Hence cosec(theta)=5/3. Checking against the original equation eliminates the extraneous root.