^{sin a sin b sin c + cos a cos b = 1 find sin c}

As nothing is given , so assume that ABC is a right isosceles triangle, with Angle a = 45, b = 45, c = 90
Hence sin a sin b sin c + cos a cos b = sin45cos45sin90 +cos45cos45 = 1
With these angles sin a sin b sin c + cos a cos b = 1 is satisfied.
Hence sinc = sin90 = 1