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`

Suppose p(x)=a_0+a_1x+a_2x^2++a_n x^ndot If |p(x)|lt=e^(x-1)-1| for all xgeq0, prove that |a_1+2a_2++n a_n|lt=1.

Prove that sinx +sin2x +sin3x =sin2x (1+2cosx) .

If a_1,a_2, a_3, a_4 be the coefficient of four consecutive terms in the expansion of (1+x)^n , then prove that: (a_1)/(a_1+a_2)+(a_3)/(a_3+a_4)=(2a_2)/(a_2+a_3)dot

If f(x)=(a_1x+b_1)^2+(a_2x+b_2)^2+...+(a_n x+b_n)^2 , then prove that (a_1b_1+a_2b_2++a_n b_n)^2lt=(a1 2+a2 2++a n2)^(b1 2+b2 2++b n2)dot

If S+a_1+a_2+a_3++a_n ,a_1 in R^+ for i=1ton , then prove that S/(S-a_1)+S/(S-a_2)++S/(S-a_n)geq(n^2)/(n-1),AAngeq2

If x lt 0 , then prove that cos^(-1) x = pi - sin^(-1) sqrt(1 - x^(2))

(cosx)/((1-sinx)(2-sinx)) [Hint : Put sin x = t]

If 1 + sin x + sin^2 x + sin ^3 x + ……. to oo = 4 +2 sqrt(3) , 0 lt x lt pi then x=

If a_1, a_2, ,a_n >0, then prove that (a_1)/(a_2)+(a_2)/(a_3)+(a_3)/(a_4)++(a_(n-1))/(a_n)+(a_n)/(a_1)> n

If a_1,a_2,a_3 ,….""a_n is an aritmetic progression with common difference d. prove that tan [ tan ^(-1) ((d)/( 1+ a_1a_2))+ tan^(-1) ((d)/(1+a_2a_3))+…+((d)/(1+a_n a_( n-1))) ] = (a_n -a_1)/(1 +a_1 a_n )