Question

sin^2 72°-sin^2 60°=√5-1/8​

Answer 1

Answer:

Step-by-step explanation:

Note that

sin(72∘)=cos(90∘−72∘)

sin(72∘)=cos(18∘)

cos(18∘)=cos(36∘/2)

For the half angle formula,

cos2(x∘)=12[1+cos(2x∘)]

or cos2(18∘)=12[1+cos(36∘)]

The value of cos(36∘) is very well known, that is:

cos(36∘)=(1+√5)/4

You can see the result of the derivation here:

Now,

cos2(18∘)=12[1+1+5√4]

or sin2(72∘)=12[1+1+5√4]

since

cos2(18∘)=sin2(72∘)

sin2(60∘)=sin(60∘)⋅sin(60∘)

= √3/2⋅√3/2

= 3/4

Now,

sin2(72∘)−sin2(60∘)

= 1/2[1 + (1+√5)/4]−3/4

= 12[5+5√4]−34

= 5+5√8−34

= (5+√5-6)/8

= (√5−1)/8

Hope this helped you

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Step-by-step explanation: