Prove that: `|1 1 1 1 1+x1 1 1 1+y|=x y` .

Prove that: |(1, 1, 1),( 1, 1+x,1) ,(1, 1, 1+y)|=x y .

underset Prove that ((x ^ (- 1) + y ^ (- 1)) / (x ^ (- 1))) ^ (- 1) + ((x ^ (- 1) -y ^ (- 1) ) / (x ^ (- 1))) ^ (- 1) = (2y ^ (2)) / (y ^ (2) -x ^ (2))

Prove that sin ^ (- 1) x + cos ^ (- 1) y = (tan ^ (- 1) (xy + sqrt ((1-x ^ (2)) (1-y ^ (2)))) ) / (y sqrt (1-x ^ (2)) - x sqrt (1-y ^ (2)))

If sin ^ (- 1) x + sin ^ (- 1) y = (pi) / (2), then prove that sin ^ (- 1) x = cos ^ (- 1) y

If y = x^x , prove that y_2 - y^-1 (y_1)^2 - yx^-1 = 0

prove that . Tan ^(-1) x- tan ^(-1 )y = cos ^(-1) (1+xy)/(sqrt((1+x^(2))(1+y^(2)))).

If y=e^x , Prove that , xy_2-(2x-1)y_1+(x-1)y=0 .

If A(x_1, y_1), B (x_2, y_2) and C (x_3, y_3) are the vertices of a DeltaABC and (x, y) be a point on the internal bisector of angle A , then prove that : b|(x,y,1),(x_1, y_1, 1), (x_2, y_2, 1)|+c|(x, y,1), (x_1, y_1, 1), (x_3, y_3, 1)|=0 where AC = b and AB=c .

If x > 0, y >0, z > 0 and x +y + z = 1 , prove that (1+x)(1+y)(1+z)ge 8(1-x) (1-y) (1-z)

If x sqrt ( 1+ y) + y sqrt( 1+x) =0 , prove that (dy)/( dx) = - (1)/( (1+x)^2) .