Question

Find the value of sin6° * sin42° * sin66° * sin102° ?





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Answer 1

Solution :-

→ sin 6° * sin 42° * sin 66° * sin 102°

solving by making pair ,

→ (sin 6° * sin 66°) * ( sin 102° * sin 42°)

using :-

• sin A * sin B = (1/2) * cos(A - B) - cos(A + B)
• cos (-A) = cos A

→ [(1/2) * {cos (6 - 66) - cos (6 + 66)}] * [(1/2) * {cos (102 - 42) - cos (102 + 42)}]

→ [(1/2) * {cos 60° - Cos 72°}] * [(1/2) * {cos 60° - Cos 144°}]

→ [(1/2) * (1/2 - cos 72°)] * [(1/2) * (1/2 - cos 144°)]

→ [(1/2) * (1/2) * (1 - 2•cos 72°)] * [(1/2) * (1/2) * [1 - 2•cos 144°]

→ (1/16)[(1 - 2•cos 72°) * (1 - 2•cos 144°)]

→ (1/16)[1 - 2•cos 144° - 2•cos 72° + 4* cos 72° * cos 144°]

→ (1/16)[1 - 2•cos 144° - 2•cos 72° + 2(2 * cos 72° * cos 144°]

using :-

• cos A * cos B = (1/2) * cos(A - B) + cos(A + B)

→ (1/16)[1 - 2•cos 144° - 2•cos 72° + 2{cos(72 - 144) + cos(72 + 144)}]

→ (1/16)[1 - 2•cos 144° - 2•cos 72° + 2{cos 72° + cos 216°}]

using :-

• cos (180 - θ) = - cos θ => cos 144° = - cos 36°
• cos (180 + θ) = - cos θ => cos 216° = - cos 36°

→ (1/16)[1 + 2•cos 36° - 2•cos 72° + 2•cos 72° - 2•cos 36°]

→ (1/16) (Ans.)

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Answer 2

${\underline{\underline{\underline{\textbf{\red{hey mate \: here \: is \: your \: answer}}}}}}$

$L.H.S. = 21[(2 sin ∘ sin 66∘)(2 sin 42∘ sin 78∘)]</p><p>=41 [(cos 60∘ - cos 72∘)(cos 36∘ -cos 120∘)]</p><p>=41[(21−45−1)(45+121)]</p><p>=4143−5.43+5=641.(9−5)=161.</p><p></p><p>$