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1)sin 12 X SIN 48 X SIN 54 =____________
1. 1/16
2. 1/32
3. 1/8
4. 1/4
one year ago

Dear Krisha

Let the expression be 'x'.

x = sin(12)sin(48)sin(54)
= [sin(12)sin(48)]sin(54)
= sin(54) * [cos(48 - 12) - cos(48 + 12)]/2
= sin(54) * [cos(36) - cos(60)]/2
= sin(54) * [cos(36) - 1/2]/2
= cos(36) * [cos(36)/2 - 1/4]
= cos²(36)/2 - cos(36)/4.

cos(36) = (1 + √5)/4.

x = cos²(36)/2 - cos(36)/4
= [(1 + √5)/4]²/2 - [(1 + √5)/4]/4
= (1 + √5)²/32 - (1 + √5)/16
= (6 + 2√5)/32 - (1 + √5)/16
= (6 + 2√5)/32 - (2 + 2√5)/32
= 4/32
= 1/8

Regards
Arun (askIITians forum expert)
one year ago
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