If `|a_1sinx+a_2sin2x++a_nsinn x|lt=|sinx|` for `x in R ,` then prove that `|a_1+2a_1+3a+3+n a_n|lt=1`

If a_1,a_2,a_3,...,a_n be in AP whose common difference is d then prove that sum_(i=1)^n a_ia_(i+1)=n{a_1^2+na_1d+(n^2-1)/3 d^2} .

If (a_1+ib_1)(a_2+ib_2)...(a_n+ib_n) = x+iy , prove that : (a_1^2+b_1^2)(a_2^2+b_2^2)...(a_n^2+b_n^2)=x^2+y^2 .

If (a_1+ib_1)(a_2+ib_2)...(a_n+ib_n) = x+iy , prove that : tan^-1frac(b_1)(a_1)+tan^-1frac(b_2)(a_2)+...+tan^-1frac(b_n)(a_n)=tan^-1fracyx .

If a_1,a_2,a_3,.....,a_n are in AP, prove that 1/(a_1a_2)+1/(a_2a_3)+1/(a_3a_4)+...+1/(a_(n-1)a_n)=(n-1)/(a_1a_n) .

If 0 lt x_(1) lt x_(2) lt x_(3) lt pi then prove that sin ((x_(1) + x_(2) + x_(3))/(3)) gt (sin x_(1) + sin x_(2) + sin x_(3))/(3) . Hence or otherwise prove that if A, B, C are angles of triangle then maximum value of sinA + sin B + sin C is (3sqrt3)/(2)

Solve |sin 3x + sinx | + |sin 3x - sinx |= sqrt3, - (pi)/(2) lt x lt (pi)/(2).