Question

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Prove That Cos 20 Cos40 Cos60 Cos80 1/ 16? explain step by step pliz..

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Answer 1

Cos 20° · Cos40° · Cos60° . Cos80° = 1/16

LHS = Cos 20° · Cos40° · Cos60° . Cos80°

We know that Cos60° = 1/2

LHS = Cos 20° · Cos40° · 1/2 . Cos80°

Multiply and divide the equation by 2

LHS = 1/4 (2. Cos 20° · Cos40° · Cos80°)

We know the formula 2 cosa cosb= cos(a+b) + cos(a-b)

LHS = 1/4 [Cos(20+80)+ Cos(20-80)] . Cos40

LHS = 1/4 [Cos(-60)+ Cos(100)] Cos40

LHS = 1/4 [1/2 + cos100] Cos40

LHS = 1/8 Cos40+ 1/4 (Cos40 . Cos100)

Multiply and divide the equation by 2

LHS = 2/2 (1/8 Cos 40) + 1/8(2 Cos40 Cos100)

We know the formula

2cosa cosb= cos(a+b) cos(a-b)

LHS = 1/8 Cos40+ 1/8 [Cos 140 + Cos (-60)]

LHS = 1/8 Cos 40+ 1/8 Cos 140 + 1/16

Since Cos 60= 1/2

LHS = 1/8 (Cos 40 + Cos 140) + 1/16

LHS = 1/8 [2 Cos 90 Cos (-50)] + 1/16

LHS = Cos 90

Cos 90 = 0

LHS = 1/16

LHS = RHS

Answer 2

L.H.S.

=(cos20°.cos40°)cos60°.cos80°

=1/2[cos(20° + 40°) + cos(20° – 40°)]×1/2×cos80°

=1/4[cos60° + cos(-20°)]cos80°

=1/4[cos60°cos80° + cos20°cos80°]

=1/4[1/2cos80° + 1/2{cos(20° + 80°) + cos(20° – 80°)}]

=1/8[cos80° + {cos100° + cos(-60°)}]

=1/8[cos80° + cos100° + cos60°]

=1/8[cos80° +cos(180° – 80°) +cos60°]

=1/8[cos80° – cos80° + cos60°]

=1/8 ×cos60°

=1/8 × 1/2

=1/16 = R.H.S

L.H.S = R.H.S = 1/16

Hence proved