IIT JEE , AIEEE Forum - Mathematics , Trignometry

loveshah

prove that -:

tan 6 * tan 42 * tan 66 * tan78 =1

each angle is in degrees.

ankur_gupta_ntse_scholar

pairing tan66 and tan6

we have

tan66*tan6= (sin66sin6+cos66cos6)/cos66cos6-1=cos60/cos66cos6-1
=1/2cos66*cos6=1/(1/2+cos72)=4/(root5+1)-1

tan72*tan48=(sin72*sin48-cos48cos72)/cos48cos72 + 1
=1/cos120+cos36 +1
=4/(root5-1) +1

i guess multiplying it will give the desired result..

Please check

rate me if u feel like doing so ..

anterr

iitkgp_bipin

tan6.tan66 = (sin6.sin66)/(cos6.cos66) = (2sin6.sin66)/(2cos6.cos66)

Now apply
2cosA.cosB = cos(A+B) + cos(A-B)
2sinA.sinB = cos(A-B) - cos(A+B)

tan6.tan66 = (cos60 - cos72)/(cos60 + cos72)

= (1/2 - sin18)/(1/2 + sin18)

Now use sin18 = (

5 - 1)/4

It will give tan6.tan66 = (3 -

5)/(

5 + 1)....................(1)

Now, tan42.tan78 = (2sin42.sin78)/(2cos42.cos78)

= (cos36 - cos120)/(cos36 + cos120)

Now use cos36 = (

5 + 1)/4 and cos120 = -1/2

You will get tan42.tan78 = (3 +

5)/(

5 - 1)................(2)

From (1) and (2) :

tan6.tan42.tan66.tan78 = (3 -

5)(3 +

5) / (

5 + 1)(

5 - 1)

= (9-5) / (5-1) = 1

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