Question

Q.12 Prove that

Cos20° cos40° cos60°cos80°=1÷
16​

Answer 1

Answer:

cos20°.cos40°.cos60°.cos80° = 1/16

cos20°.cos40°.cos60°.cos80° = 1/16L.H.S.

cos20°.cos40°.cos60°.cos80° = 1/16L.H.S.=(cos20°.cos40°)cos60°.cos80°

cos20°.cos40°.cos60°.cos80° = 1/16L.H.S.=(cos20°.cos40°)cos60°.cos80°=1/2[cos(20° + 40°) + cos(20° – 40°)]×1/2×cos80°

cos20°.cos40°.cos60°.cos80° = 1/16L.H.S.=(cos20°.cos40°)cos60°.cos80°=1/2[cos(20° + 40°) + cos(20° – 40°)]×1/2×cos80°=1/4[cos60° + cos(-20°)]cos80°

cos20°.cos40°.cos60°.cos80° = 1/16L.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°]

cos20°.cos40°.cos60°.cos80° = 1/16L.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°)}]

cos20°.cos40°.cos60°.cos80° = 1/16L.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°)}]

cos20°.cos40°.cos60°.cos80° = 1/16L.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°]

cos20°.cos40°.cos60°.cos80° = 1/16L.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°]

cos20°.cos40°.cos60°.cos80° = 1/16L.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°]

cos20°.cos40°.cos60°.cos80° = 1/16L.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°

cos20°.cos40°.cos60°.cos80° = 1/16L.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

cos20°.cos40°.cos60°.cos80° = 1/16L.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

cos20°.cos40°.cos60°.cos80° = 1/16L.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.SL.H.S = R.H.S = 1/16 Hence proved

Step-by-step explanation:

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