Question

sin 10°sin30°sin50°sin70°= 1÷16​

Answer 1

Answer:

$How can you prove that sin 10° sin 30° sin 50° sin 70° = 1/16?</p><p></p><p>sin10°=sin(90–80)°=cos80°sin⁡10°=sin⁡(90–80)°=cos⁡80°</p><p></p><p>sin50°=sin(90–40)°=cos40°sin⁡50°=sin⁡(90–40)°=cos⁡40°</p><p></p><p>sin70°=sin(90–20)°=cos20°sin⁡70°=sin⁡(90–20)°=cos⁡20°</p><p></p><p>∴sin10°sin30°sin50°sin70°=cos80°sin30°=12cos40°cos20°∴sin⁡10°sin⁡30°sin⁡50°sin⁡70°=cos⁡80°sin⁡30°⏟=12cos⁡40°cos⁡20°</p><p></p><p>=12⋅cos20°cos40°cos80°=14sin20°⋅(2sin20°cos20°)cos40°cos80°=12⋅cos⁡20°cos⁡40°cos⁡80°=14sin⁡20°⋅(2sin⁡20°cos⁡20°)cos⁡40°cos⁡80°</p><p></p><p>=18sin20°⋅(2sin40°cos40°)cos80°=18sin⁡20°⋅(2sin⁡40°cos⁡40°)cos⁡80°</p><p></p><p>=116sin20°⋅(2sin80°cos80°)=116sin⁡20°⋅(2sin⁡80°cos⁡80°)</p><p></p><p>=116sin20°⋅(sin160°)=116sin20°⋅(sin(180−20°))=116sin⁡20°⋅(sin⁡160°)=116sin⁡20°⋅(sin⁡(180−20°))</p><p></p><p>=116sin20°⋅sin20°=116=116sin⁡20°⋅sin⁡20°=116</p><p></p><p></p><p>$

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Answer 2

Answer:

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Step-by-step explanation:

We know that

i) sin 2A = 2sinAcosA

ii) sin(90-A)= cosA

iii)sin30=1/2

LHS = sin 10 sin30 sin50sin 70

=sin 10 × 1/2 × sin (90-40)× sin(90-20)

=1/2[sin 10 ×cos 40×cos 20]

We multiply with[2 cos 10/2cos 10]

=(1/2×1/2cos10)[2sin10cos10×cos20×cos40]

=(1/4cos10)[sin20cos20×cos40]

=(1/8cos10)[2sin20cos20×cos40]

=(1/8cos10)[sin40cos40]

=(1/16cos10)[2sin40co40]

=(1/16cos10)×sin80

=(1/16cos10)×sin(90-10)

=(1/16cos10)×cos10

=1/16

= RHS