Prove that sin(pi/15) = (1/4)√7 - √5 - √(30 - 6√5)?

Question

Here are the rules:

sin(pi/15) = sin(2pi/5 - pi/3) = sin(pi/6 - pi/10)

sin(a-b) = sin(a)cos(b)-sin(b)cos(a)

sin(pi/10)=cos(2pi/5) = (√5 - 1)/4
cos(pi/6)=sin(pi/3) = √3/2
cos(pi/10)=sin(2pi/5) = √[(5 + √5)/8]
sin(pi/6)=cos(pi/3) = 1/2

Good luck. I am not able to do it.

Historically, this is a key part to the solution of Van Roomen's problem.

This is the solution according to Mathworld.wolfram.com:

Checkout:
http://mathworld.wolfram.com/TrigonometryAnglesPi15.htm

I will also accept a proof that the equation is wrong.

This is the solution according to Mathworld.wolfram.com:

Checkout:
http://mathworld.wolfram.com/TrigonometryAnglesPi15.html

I will also accept a proof that the equation is wrong.

To be clear the parentheses should be around the entire equation so that is:

(1/4)sqrt{7 - sqrt(5) - sqrt[30 - 6*sqrt(5)]}

It should be:

(1/4)√{7 - √5 - √(30 - 6√5)}

So, it is not a negative number. Sorry that this wasn't clear.

Also, the two numbers are equal so there won't be a proof that the equation is wrong.

Answer 1

No. But I can prove that it does not equal that expression. sin (π/15) is a positive number. Whereas (1/4)√7 - √5 - √(30 - 6√5) is a negative number. So they cannot be equal.

Edit: "Sorry this wasn't clear"? What planet do you live on where (1/4)√7 - √5 - √(30 - 6√5) could possibly be interpreted to mean (1/4)√(7 - √5 - √(30 - 6√5))?

Anyway, on the page you link to, they derive cos (/15) in the form 1/8 * (√(30+6√5) + √5 - 1). The particular form of sin (π/15) is derived from that. Observe:

cos (π/15) = 1/8 * (√(30+6√5) + √5 - 1)
cos² (π/15) = 1/64 * (30+6√5 + 5 + 1 + 2√(30+6√5)√5 - 2√(30+6√5) - 2√5)
cos² (π/15) = 1/64 * (36 + 4√5 + 2√(30+6√5)√5 - 2√(30+6√5))
sin² (π/15) = 1-cos² (π/15) = 1/64 * (28 - 4√5 - 2√(30+6√5)√5 + 2√(30+6√5))
sin² (π/15) = 1/64 * (28 - 4√5 - 2√(30+6√5)√5 + 2√(30+6√5))
sin² (π/15) = 1/64 * (28 - 4√5 - (2√5-2)√(30+6√5))
sin² (π/15) = 1/64 * (28 - 4√5 - √(2√5-2)²√(30+6√5))
sin² (π/15) = 1/64 * (28 - 4√5 - √(24-8√5)√(30+6√5))
sin² (π/15) = 1/64 * (28 - 4√5 - √(720-240√5+144√5-240))
sin² (π/15) = 1/64 * (28 - 4√5 - √(480-96√5))
sin² (π/15) = 1/64 * (28 - 4√5 - √16√(30-6√5))
sin² (π/15) = 1/64 * (28 - 4√5 - 4√(30-6√5))

sin (π/15) = 1/4 * √(7 - √5 - √(30-6√5)). Q.E.D.

Answer 2

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Answer 3

cos(π/3) = 1/2
sin(π/3) = (√3)/2
cos(2π/5) = (√5 - 1)/4
sin(2π/5) = [√(10 + 2√5)]/4

sin(π/15) = sin(2π/5 - π/3)
= (sin(2π/5))(cos(π/3)) - (cos(2π/5))(sin(π/3))
= {[√(10 + 2√5)]/4}(1/2) - {(√5 - 1)/4}{(√3)/2}
= {[√(10 + 2√5)]/8} - {(√15 - √3)/8}
= {√(10 + 2√5) - √15 + √3}/8