If `sum_(n=1)^nalpha_n=a n^2+b n ,w h e r ea ,b` are constants and `alpha_1,alpha_2,alpha_3 in {1,2,3,......,9}a n d25alpha_1,37alpha_2,49alpha_3` be three digit number, then prove that `|alpha_1alpha_2alpha_3 5 7 9 25alpha_1 37alpha_2 49alpha_3|=0`

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

If 0 lt alpha lt (pi)/(3) , then prove that alpha (sec alpha) lt (2pi)/(3).

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 the roots of equation x^3+a x^2+b=0a r ealpha_1,alpha_2 and alpha_3(a ,b!=0) , then find the equation whose roots are (alpha_1alpha_2+alpha_2alpha_3)/(alpha_1alpha_2alpha_3),(alpha_2alpha_3+alpha_3alpha_1)/(alpha_1alpha_2alpha_3),(alpha_1alpha_3+alpha_1alpha_2)/(alpha_1alpha_2alpha_3)

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)) .

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

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

Prove that [1+cot alpha-sec(alpha+pi/2)] [1+cot alpha+sec (alpha+pi/2)]=2 cot alpha .

Prove that |a_1alpha_1+b_1beta_1a_1alpha_2+b_2beta_2a_1alpha_3+b_1beta_3a_2alpha_1+b_2beta_1a_2alpha_2+b_2beta_2a_2alpha_3+b_2beta_3a_ 3alpha_1+b_3beta_1a_3alpha _2+b_3beta_2a_3alpha_3+b_3beta_3|=0