Prove that :
`sin (y + z-x) + sin (z + x-y) + sin (x +y-z)-sin (x+y+z)` =`4 sin x sin y sin z`.

Prove that : sin x sin y sin (x - y) + sin y sin z sin (y- z)+ sin z sin x sin (z - x) + sin (x - y) sin (y-z)sin (z- x) = 0 .

Prove that : tan (x-y)+tan (y -z) + tan (z-x)= tan (x-y)tan (y -z) tan (z-x) .

If (sin x)^Y = (sin y)^x , find dy/dx :

Prove that: (cos x + cos y)^2 + (sin x -sin y)^2 = 4 cos^2frac (x+y)(2)

Find dy/dx when (sin x)^y = (sin y)^x

If sin y = x sin (a + y), then prove that (dy)/(dx) = (sin^(2) (a + y))/(sin a) .

Prove that (sin(x+y))/(sin(x-y))=(tanx+tany)/(tanx-tany)

Statement I The system of linear equations x+(sin alpha )y+(cos alpha )z=0 x+(cos alpha ) y+(sin alpha )z=0 -x+(sin alpha )y-(cos alpha )z=0 has a not trivial solution for only one value of alpha lying between 0 and pi . Statement II |{:(sin x, cos x , cos x),( cos x , sin x , cos x) , (cos x , cos x , sin x ):}|=0 has no solution in the interval -pi//4 lt x lt pi//4 .

If sin x = y sin (x + b) , show that (dy)/(dx) = (sin b)/(sin^2 (x + b))