If (1 + i) (1 + 2i) (1 + 3i) ... (1 + ni) = `alpha + ibeta`, prove that `2.5.10.....(1+n^2)=alpha^2+beta^2`

If the points A(1, 0), (0, 1) and ( alpha , beta ) are collinear prove that alpha + beta = 1

If tan alpha=m/(m+1) and tan beta=1/(2m+1) ,prove that alpha+beta=pi/4 .

If tan^2alpha = 1 + tan^2beta ,then prove that cos^2 beta = cot^2alpha

If 2 tan beta + cot beta = tan alpha , prove that 2 tan (alpha-beta) = cot beta

Prove that 1/3ltlog_10 3lt1/2

Prove that 1/4ltlog_10 2lt1/3

If (p+1 ,1),(2p+1 ,3), (2p+2, 2p) are collinear prove that,p= 2 or -1/2

By mathematical induction prove that 1+3+5+7+…..+(2n-1)=n^2

If "tan"^(-1) (sqrt(1+x^(2))-sqrt(1-x^(2)))/(sqrt(1+x^(2))+sqrt(1-x^(2)))=alpha , then prove that x^(2) =sin 2alpha .

If A = [(3, 1),(-1, 2)] show that A^2 - 5A + 7I = 0 .