Home
Class 12
MATHS
Let alpha,beta,gamma>0a n dalpha+beta+ga...

Let `alpha,beta,gamma>0a n dalpha+beta+gamma=pi/2dot` Then prove that `sqrt(tanalphatanbeta)` + `sqrt(t a nbeta"tan"gamma)+sqrt(tanalphatangamma)lt=sqrt(3)`

Text Solution

Verified by Experts

Let `p=tan alpha beta,q=tan beta gamma "and" r=tan alpha tan gamma`
Now, `(sqrt(p)+sqrt(q)+sqrt(4))^(2)=p+q+r+2sqrt(pq)+2sqrt(qr)_2sqrt(rp)`
Also, using `GM leAM` we have
`2sqrt(pq)lep+q,2sqrt(qr)leq+r,2sqrt(rp)ler+p`
`therefore (sqrt(p)+sqrt(1)+sqrt(r))^(2)lep+q+r+p+q+q+r+r+p`
`therefore sqrt(p)+sqrt(q)+sqrt(r)lesqrt(3).sqrt(p+q+r)`
Now for `alpha+beta+gamma=(pi)/(2)`, we have
`p+q+r=tan alpha tan beta+tan tan gamma+tan alpha tan gamma=1`
`therefore sqrt(p)+sqrt(q)+sqrt(r)ltsqrt(3)`
Promotional Banner

Topper's Solved these Questions

  • TRIGONOMETRIC RATIOS AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Exercise (Single)|100 Videos

  • TRIGONOMETRIC RATIOS AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Exercise (Multiple)|22 Videos

  • TRIGONOMETRIC RATIOS AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Exercise 3.9|5 Videos

  • TRIGONOMETRIC FUNCTIONS

    CENGAGE|Exercise SINGLE CORRECT ANSWER TYPE|38 Videos

  • TRIGONOMETRIC RATIOS FOR COMPOUND, MULTIPLE, SUB-MULTIPLE ANGLES, AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Multiple Correct Answers Type|6 Videos

Similar Questions

Explore conceptually related problems

Let alpha,beta,gamma>0 and alpha+beta+gamma=(pi)/(2). Then prove that sqrt(tan alpha tan beta)+sqrt(tan beta tan gamma)+sqrt(tan alpha tan gamma)<=sqrt(3)

Let alpha,beta,gamma>0 and alpha+beta+gamma=(pi)/(2). Statement-1: |tan alpha tan beta-(a!)/(6)|+|tan beta tan gamma-(b!)/(2)|+|tan gamma tan alpha-(c!)/(3)|<=0 where n!=1.2............n, then tan alpha tan beta,tan beta tan gamma,tan gamma tan alpha=1

If alpha+beta+gamma=2pi , then show that tan.alpha/2 + tan.beta/2 + tan.gamma/2 = tan.alpha/2 tan.beta/2 tan.gamma/2 .

If alpha,beta,gamma,in(0,(pi)/(2)), then prove that (si(alpha+beta+gamma))/(sin alpha+sin beta+sin gamma)<1

If alpha+beta=(pi)/(2) and beta+gamma=alpha then tan alpha=

If alpha,beta,gamma are acute angle such that cos alpha=tan beta,cos beta=tan gamma and cos gamma=tan alpha then

If alpha+ beta + gamma=90^(@) , then tan alpha+ tan beta + tan alpha tan beta cot gamma is __________

If tan beta=(tan alpha + tan gamma)/(1+ tan alpha . tan gamma) Prove that sin 2 beta =(sin 2 alpha + sin 2 gamma)/(1+ sin 2 alpha . Sin 2 gamma)

Let alpha=som^(-1)((36)/(85)),beta=cos^(-1)((4)/(5)) and gamma=tan^(-1)((8)/(15)) then cot alpha+cot beta+cot gamma=cot alpha cot beta cot gammatan alpha tan beta+tan beta tan gamma+tan alpha tan gamma=1tan alpha+tan beta+tan gamma=tan alpha tan beta tan gammacot alpha cot beta+cot beta cot gamma+cot alpha cot gamma=1