If `0ltalphaltpi/2,` show that `alpha- alpha^3/3ltsinalpha`

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

If 0ltalphaltpi/2 and sinalpha+cosalpha+tanalpha+cotalpha+secalpha+cosecalpha="7, then prove that sin2alpha is a root of the equation x^2-44x-36=0.

If 0ltalphaltpi/4 then the range of cosec 2alpha-cot 2alpha is (A) (0,1) (B) [1,oo) (C) RR (D) [0,oo)

if tan alpha = 2tanbeta show that (sin(alpha+beta))/(sin(alpha-beta)) = 3

(a) show that alpha^2 + beta^2 + alpha beta =0

Given that alpha and beta are the roots of the equation x^(2)=7x+4 , (i) show that alpha^(3)=53alpha+28 (ii) find the value of (alpha)/(beta)+(beta)/(alpha) .

Let A=[(0,-tan(alpha/2)),(tan(alpha/2),0)] and I be the identity matrix of order 2. Show that I+A=(I-A)[cosalpha-sin alpha sin alpha cosalpha] .

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 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))=nalpha_(1)^(n-1)+nalpha

If 0ltalpha,betaltpi and cos alpha+cos beta -cos(alpha+beta)=(3)/(2) , then the value of sqrt3 sin alpha+cos alpha is equal to