help@askiitians.com

+91-93193 64476

Trigonometry> Let α and β be non–zero real numbers such...

question mark

Let α and β be non–zero real numbers such that 2(cosβ – cosα) + cosα cosβ = 1. Then which of the following is/are true?

Umang Kumar , 7 Years ago

Grade 12

1 Answers

Umang Kumar

2(cosb-cosa)+cosa.cosb=1

2cosb-2cosa+cosa.cosb=1

cosb(2+cosa)-(2cosa+1)=0

$cos\beta =\frac{2cos\alpha +1}{2+cos\alpha }$

Now converting $cos\beta =\frac{1-tan^{2}\frac{\beta }{2}}{1+tan^{2}\frac{\beta }{2}}$

$cos\alpha =\frac{1-tan^{2}\frac{\alpha }{2}}{1+tan^{2}\frac{\alpha }{2}}$

now equating cosa and cosb in

$cos\beta =\frac{2cos\alpha +1}{2+cos\alpha }$

we get

$\frac{1-tan^{2}\frac{\beta }{2}}{1+tan^{2}\frac{\beta }{2}}=\frac{3-tan^{2}\frac{\alpha }{2}}{3+tan^{2}\frac{\alpha }{2}}$

After applying Componendo and dividendo method each side we get:

$\frac{2}{-2tan^{2}\frac{\beta }{2}}=\frac{6}{-2tan^{2}\frac{\alpha }{2}}$

After solving this we get:

$\frac{1}{tan^{2}\frac{\beta }{2}}=\frac{3}{tan^{2}\frac{\alpha }{2}}$

After cross multiplying we get:

$tan^{2\frac{\alpha }{2}}=3tan^{2\frac{\beta }{2}}$

Square root both side:

$tan\frac{\alpha }{2}=\pm \sqrt{3}tan\frac{\beta }{2}$

hence the answers are:-

1. $tan\frac{\alpha }{2}+\sqrt{3}tan\frac{\beta }{2}=0$

2. $tan\frac{\alpha }{2}-\sqrt{3}tan\frac{\beta }{2}=0$

Last Activity: 7 Years ago

LIVE ONLINE CLASSES

Prepraring for the competition made easy just by live online class.

Full Live Access

Study Material

Live Doubts Solving

Daily Class Assignments

In ABC, if cot A+cot B+cot C = cosecA cosecB cosecC, then prove that ABC is right angle triangle.

trigonometry

1 Answer Available

Last Activity: 3 Years ago

Find the general solution of equation tan3∂=cot2∂
For all angles between -720 and +720

trigonometry

1 Answer Available

Last Activity: 4 Years ago

To prove : Cosec(45 degree - A)Cosec(45 degree + A) = 2SecA

trigonometry

1 Answer Available

Last Activity: 4 Years ago

Derive a formula for the area of the triangle, given b, A, B, and C. That is, derive the formula: (b^2sinAsinC)/(2sinB)

trigonometry

1 Answer Available

Last Activity: 4 Years ago

WHAT IS THE LIMIT WHEN X TOWARDS TO 0 OF sin6x/5x ? lim𝑥→0 𝑠𝑖𝑛6𝑥/5𝑥

trigonometry

1 Answer Available

Last Activity: 4 Years ago

View all Questions