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

If cot^(-1)(sqrt(cosalpha))-tan^(-1)(sqrt(cosalpha))=x , then sinx is (a) tan^2(alpha/2) (b) cot^2(alpha/2) (c) tan^2alpha (d) cotalpha/2

Prove that |{: (1,alpha,alpha^2), (alpha,alpha^2, 1),(alpha^2, 1,alpha) :}|=-(1-alpha^3)^2

If Delta_(1) is the area of the triangle with vertices (0, 0), (a tan alpha, b cot alpha), (a sin alpha, b cos alpha), Delta_(2) is the area of the triangle with vertices (a sec^(2) alpha, b cos ec^(2) alpha), (a + a sin^(2)alpha, b + b cos^(2)alpha) and Delta_(3) is the area of the triangle with vertices (0,0), (a tan alpha, -b cot alpha), (a sin alpha, b cos alpha) . Show that there is no value of alpha for which Delta_(1), Delta_(2) and Delta_(3) are in GP.

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

Prove that 2tanbeta+cotbeta=tanalphaimplies2tan(alpha-beta)=cotbeta .

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 .

Which of the following values of alpha satisfying the equation |(1+alpha)^2(1+2alpha)^2(1+3alpha)^2(2+alpha)^2(2+2alpha)^2(2+3alpha)^2(3+alpha)^2(3+2alpha)^2(3+3alpha)^2|=-648alpha? -4 b. 9 c. -9 d. 4

If A(alpha)=[{:(cosalpha,sinalpha),(-sinalpha,cosalpha):}] then prove that A(alpha)A(-alpha)=I .

If msintheta=nsin(theta+2alpha) , then prove that tan(theta+alpha)cotalpha=(m+n)/(m-n) .

If sin (3alpha) /cos(2alpha) < 0 If alpha lies in