If `1,alpha_1,alpha_2, ,alpha_(n-1)` are the `n t h` roots of unity, prove that `(1-alpha_1)(1-alpha_2)(1-alpha_(n-1))=ndot` Deduce that `sinpi/nsin(2pi)/n sin((n-1)pi)/n=n/(2^(n-1))`

If alpha_1, ,alpha_2, ,alpha_n are the roots of equation x^n+n a x-b=0, show that (alpha_1-alpha_2)(alpha_1-alpha_3).......(alpha_1-alpha_n)=n(alpha_1^(n-1)+a)

If n ge3 and 1,alpha_1, alpha_2, alpha_3....alpha_(n-1) are the , nth roots of unity then find the value of sum_(1 le i lt j le (n-1)) (alpha_i alpha_j )

If y = (2 sin alpha )/(1 + cos alpha + sin alpha) then, prove that (1 - cos alpha + sin alpha)/(1 + sin alpha) = y .

If alpha_(1), alpha_(2), alpha_(3), alpha_(4) are the roots of the equation x^(4)+(2-sqrt(3))x^(2)+2+sqrt(3)=0 , then the value of (1-alpha_(1))(1-alpha_(2))(1-alpha_(3))(1-alpha_(4)) is

If b_1, b_2 b_n are the nth roots of unity, then prove that ^n C_1dotb_1+^n C_2dotb_2++^n C_ndotb_n=b_1/b_2{(1+b_2)^n-1}^dot

Prove that sin4 alpha=4tan alpha (1- tan^(2)alpha)/((1+tan^2 alpha)^2

If alpha is the nth root of unity, then 1+2alpha+3alpha^2+ ton terms equal to a. (-n)/((1-alpha)^2) b. (-n)/(1-alpha) c. (-2n)/(1-alpha)"" d. (-2n)/((1-alpha)^2)

If the roots of equation x^(3) + ax^(2) + b = 0 are alpha _(1), alpha_(2), and alpha_(3) (a , b ne 0) . Then find the equation whose roots are (alpha_(1)alpha_(2)+alpha_(2)alpha_(3))/(alpha_(1)alpha_(2)alpha_(3)), (alpha_(2)alpha_(3)+alpha_(3)alpha_(1))/(alpha_(1)alpha_(2)alpha_(3)), (alpha_(1)alpha_(3)+alpha_(1)alpha_(2))/(alpha_(1)alpha_(2)alpha_(3)) .

If sum_(r=0)^n{a_r(x-alpha+2)^r-b_r(alpha-x-1)^r}=0, then prove that b_n-(-1)^n a_n=0.

Prove that sin 4alpha = 4 tan alpha (1 - tan^(2) alpha)/((1 + tan^(2) alpha)^(2))