A right angle is divided into three positive parts `alpha,betaa n dgammadot` Prove that for all possible divisions `t a nalpha+tanbeta+tangamma>1+tanalphatanbetatangammadot`

If a right angle be divided into three parts alpha, beta and gamma , prove that cot alpha = (tan beta + tan gamma)/(1-tan beta tan gamma) .

Let alpha,betaa n dgamma be some angles in the first quadrant satisfying tan(alpha+beta)=(15)/8a n dcos e cgamma=(17)/8, then which of the following hold(s) good? (a) alpha+beta+gamma=pi (b) cotalpha+cotbeta+cotgamma=cotalphacotbetacotgamma (c) tanalpha+tanbeta+tangamma=tanalphatanbetatangamma (d) tanalphatanbeta+tanbetatangamma+tangammatanalpha=1

If the angle of elevation of a cloud from a point h metres above a lake is alpha and the angle of depression of its reflection in the take is beta , prove that the height of the cloud is (h(tanbeta+t a nalpha))/(tanbeta-t a nalpha)

If the angle of elevation of a cloud from a point h metres above a lake is alpha and the angle of depression of its reflection in the lake is beta , prove that the height of the cloud is (h(tanbeta+t a nalpha))/(tanbeta-t a nalpha)

If each of alpha, betaa n dgamma is a positive acute angle such that sin(alpha+beta-gamma) =1/2 , cos(beta+gamma-alpha)=1/2 and tan(gamma+alpha-beta)=1, find the values of alpha,betaa n dgammadot

If alphaa n dbeta are acute angles such that t a nalpha=m/(m+1)a n dt a nbeta=1/(2m+1) , prove that alpha+beta=pi/4 .

Using binomial theorem, prove that 5^(4n)+52n-1 is divisible by 676 for all positive integers n.

The base of a triangle is divided into three equal parts. If theta_(1), theta_(2), theta_(3) be thw angles subtended by these parts at the vertex, then prove that (cot theta_(1) +cot theta_(2) ) (cot theta_(2)+ cot theta_(3))=4 cosec ^(2) theta _(2)

The base of a triangle is divided into three equal parts. If t_1, t_2,t_3 are the tangents of the angles subtended by these parts at the opposite vertex, prove that (1/(t_1)+1/(t_2))(1/(t_2)+1/(t_3))=4(1+1/(t2 2))dot

If the normals to the parabola y^2=4a x at P meets the curve again at Q and if P Q and the normal at Q make angle alpha and beta , respectively, with the x-axis, then t a nalpha(tanalpha+tanbeta) has the value equal to 0 (b) -2 (c) -1/2 (d) -1