prove that : `tan(alpha)+2 tan(2alpha) +4(tan4alpha)+8cot(8alpha) = cot(alpha)`

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

If tan theta=(sin alpha- cos alpha)/(sin alpha+cos alpha) , then:

tan theta + 2 tan 2 theta + 4 tan 4 theta + 8 cot 8 theta = cot theta.

Using Principle of Mathematical Induction, prove that : cos alpha cos 2 alpha cos 4 alpha ............. cos (2^(n-1) alpha)= (sin (2^n alpha))/(2^n sin alpha) for all n in N.

If tan beta= (Q sin alpha)/(P+Q cos alpha) , prove that : tan (alpha-beta)= (P sin alpha)/(Q+P cos alpha) .

If sin 3 alpha =4 sin alpha sin (x+alpha ) sin(x-alpha ) , then

If alpha, beta are the roots of the quadratic equation x^2 + bx - c = 0 , the equation whose roots are b and c , is a. x^(2)+alpha x- beta=0 b. x^(2)-[(alpha +beta)+alpha beta]x-alpha beta( alpha+beta)=0 c. x^(2)+[(alpha + beta)+alpha beta]x+alpha beta(alpha + beta)=0 d. x^(2)+[(alpha +beta)+alpha beta)]x -alpha beta(alpha +beta)=0

If tan(2alpha+beta)=x & tan(alpha+2beta)=y , then [ tan3(alpha+beta)]. [ tan(alpha-beta)] is equal to (wherever defined)

Show that the lines (1-cos theta tan alpha) y^2 -(2cos theta+sin ^2 theta tan alpha )xy +cos theta (cos theta+tan alpha )x^2=0 include an angle alpha between them .

A_(alpha )= [(cos alpha, sin alpha),(- sin alpha, cos alpha)] then prove that A_(alpha . A_(beta) = A_(alpha + beta)