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)

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+tanbeta+tangamma=tanalphatanbetatangamma (c)tanalpha+tanbeta+tangamma=tanalphatanbetatangamma (d)tanalphatanbeta+tanbetatangamma+tangammatanalpha=1

A cylindrical container is divided into three parts by two tight fitting pistons. The pistons are connected by a spring. The region between the pistons is vacuum and the other two parts have same number of moles of an ideal gas. Initially, both the gas chambers are at temperature T_(0) and the spring is compressed by 1 m. Length of both gas chambers is 1 m in this position. Now the temperature of the left and right chambers are raised to (4T_(0))/(3) and (5T_(0))/(3) respectively. Find the final compression in the spring in equilibrium. Assume that the pistons slide without friction

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)

If P=x+(1)/(x) and P(tan alpha)=2,P(tan beta)=2sqrt(2),P(cot gamma)=4 where alpha,beta,gamma are acute angles and sum of all possible distinct values of alpha,beta and gamma is given by (m pi)/(n) where m and n are coprime,then find the value of (m+n)

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)/(t_(2)^(2)))

An insulated cylindrical vessel is divided into three identical parts by two partitions 1 and 2. The left part contains O_(2) gas, the middle part has N_(2) and the third chamber has vacuum. The average molecular speed in oxygen chamber is V_(0) and that in nitrogen chamber is sqrt((8)/(7)) V_(0) . Pressure of the gases in two chambers is same. Partition 1 is removed and the gases are allowed to mix. Now the stopper holding the partition 2 is removed and it slides to the right wall of the container, so that the mixture of gases occupy the entire volume of the container. Find the average speed of O_(2) molecules now.

The bae of a triangle is divided into three equal parts. If t_(1), t_(2), t_(3) be the tangent sof the angles subtended by these parts at the opposite vertex, prove that : ((1)/(t_(1))+ (1)/(t_(2)))((1)/(t _(1))+(1)/(t _(3)))=4(1+(1)/(t_(1)^(2)))