Question

prove that : tan20 tan40 tan80=tan 60

Answer 1

HELLO DEAR,

TO PROVE :-

tan20 * tan40 * tan80 = tan60

LH.S = 2sin20sin40sin80/2cos20cos40cos80

= [{cos(20 - 40) - cos(20 + 40)} sin80]/[{cos(20 + 40) + cos(20 - 40)}cos80]

= {(cos20 - cos60)sin80} / {(cos60 + cos20)cos80}

= {2cos20sin80 -2(1/2)sin80} / {2(1/2)cos80

+ 2cos20cos80}

divide and multiply by "2"

›››››››››››››››››››››››››[as cos60° = 1/2]

= {sin(20 + 80) - sin(20 - 80) - sin80} / {cos80 + cos(20 + 80) + cos(20 - 80)}

= (sin100 + sin60 - sin80) / (cos80 + cos100 + cos60)

= {2cos(100 + 80)/2 sin(100 - 80)/2 + sin60} / {2cos(100 + 80)/2 cos(100 - 80)/2 + cos60}

= (2cos90sin10 + sin60) / (2cos90cos10 + cos60)

›››››››››››››››››››››››››[cos(90) = 0]

= sin60 / cos60

= tan60

HENCE, tan20 tan40 tan80 = tan 60

I HOPE ITS HELP YOU DEAR,

THANKS

Answer 2

Following is the proof;

tan20 * tan40 * tan80 = tan60

LH.S = 2sin20sin40sin80/2cos20cos40cos80

= [{cos(20 - 40) - cos(20 + 40)} sin80]/[{cos(20 + 40) + cos(20 - 40)}cos80]

= {(cos20 - cos60)sin80} / {(cos60 + cos20)cos80}

= {2cos20sin80 -2(1/2)sin80} / {2(1/2)cos80+ 2cos20cos80}

divide and multiply by "2"

[as cos60° = 1/2]

= {sin(20 + 80) - sin(20 - 80) - sin80} / {cos80 + cos(20 + 80) + cos(20 - 80)}

= (sin100 + sin60 - sin80) / (cos80 + cos100 + cos60)

= {2cos(100 + 80)/2 sin(100 - 80)/2 + sin60} / {2cos(100 + 80)/2 cos(100 - 80)/2 + cos60}

= (2cos90sin10 + sin60) / (2cos90cos10 + cos60)

[cos(90) = 0]

= sin60 / cos60

= tan60

Therefore, tan20 tan40 tan80 = tan 60

I hope you are satisfied with the answer.

Thanks