# 4 Sine Pi over 10 by Cosine Pi over 5/Proof 2

## Theorem

$4 \sin \dfrac \pi {10} \cos \dfrac \pi 5 = 1$

## Proof

First, we must rewrite the left hand side so that it reaches a form that is able to be substituted for what we determine is true through identities.

 $\displaystyle 4 \sin \dfrac \pi {10} \cos \dfrac \pi 5$ $=$ $\displaystyle 1$ Given $\displaystyle 4 \sin \dfrac \pi {10} \cos \dfrac {2 \pi} {10}$ $=$ $\displaystyle 1$ multiplying the angle inside the cosine by $\dfrac 2 2$ $\displaystyle 4 \sin \dfrac \pi {10} \paren { 2 \paren { \cos \dfrac \pi {10} }^2 - 1}$ $=$ $\displaystyle 1$ Double-angle formula (cosine) $\displaystyle 4 \sin \paren { \dfrac 1 2 \dfrac \pi {5} } \paren { 2 \paren { \cos \dfrac \pi {10} }^2 - 1}$ $=$ $\displaystyle 1$ Factoring a $\dfrac 1 2$ out of $\dfrac \pi {10}$ $\displaystyle 4 \sqrt { \dfrac { 1 - \paren { 2 \paren { \cos \dfrac \pi {10} }^2 - 1} } 2 } \paren { 2 \paren { \cos \dfrac \pi {10} }^2 - 1}$ $=$ $\displaystyle 1$ $\sin \dfrac \theta 2 = \sqrt { \dfrac { 1 - \cos \theta } 2 }$ $\displaystyle 4 \sqrt { 1 - \cos^2 \dfrac \pi {10} } \paren { 2 \paren { \cos \dfrac \pi {10} }^2 - 1}$ $=$ $\displaystyle 1$ Trivial simplification

Now that we have reduced the expression to just one trigonometric function of a single angle, we can use the fact that a multiple of the angle is $\dfrac \pi 2$, whose cosine is $0$, in order to solve for $\cos \dfrac \pi {10}$.

 $\displaystyle \cos \dfrac \pi 2$ $=$ $\displaystyle 0$ (New equation) Starting with $5 \theta$ and its value $\displaystyle \cos \paren { \dfrac \pi 5 + \dfrac {3\pi} {10} }$ $=$ $\displaystyle 0$ Rewriting $5 \theta$ as $2 \theta + 3 \theta$ $\displaystyle \cos \dfrac \pi 5 \cos \dfrac {3\pi} {10} - \sin \dfrac \pi 5 \sin \dfrac {3\pi} {10}$ $=$ $\displaystyle 0$ Double-angle formula (cosine) $\displaystyle \paren { 2 \cos^2 \dfrac \pi {10} - 1 } \paren { 4 \cos^3 \dfrac {3\pi} {10} - 3 \cos \dfrac \pi {10} } - \paren { 2 \sin \dfrac \pi {10} \cos \dfrac \pi {10} } \paren { 3 \sin \dfrac \pi {10} - 4 \sin^3 \dfrac \pi {10} }$ $=$ $\displaystyle 0$ Multiple-angle formulas (cosine, sine) $\displaystyle \paren { 8 \cos^5 \dfrac \pi {10} - 10 \cos^3 \dfrac \pi {10} + 3 \cos \dfrac \pi {10} } - \paren { 6 \sin^2 \dfrac \pi {10} \cos \dfrac \pi {10} - 8 \sin^4 \dfrac \pi {10} \cos \dfrac \pi {10} }$ $=$ $\displaystyle 0$ Multiplication $\displaystyle \paren { 8 \cos^5 \dfrac \pi {10} - 10 \cos^3 \dfrac \pi {10} + 3 \cos \dfrac \pi {10} } - \paren { 6 \cos \dfrac \pi {10} - 6 \cos^3 \dfrac \pi {10} - 8 \cos \dfrac \pi {10} + 16 \cos^3 \dfrac \pi {10} - 8 \cos^5 \dfrac \pi {10} }$ $=$ $\displaystyle 0$ Pythagorean Identity $\displaystyle 16 \cos^5 \dfrac \pi {10} - 20 \cos^3 \dfrac \pi {10} + 5 \cos \dfrac \pi {10}$ $=$ $\displaystyle 0$ Trivial simplification

Having realized that the above process is the same as the derivation of the expansion formula for $\cos {5 \theta}$, we can generalize this expression and then plug in $\dfrac \pi {10}$.

 $\displaystyle 16u^5 - 20u^3 + 5u$ $=$ $\displaystyle 0$ $u = \cos \theta$ $\displaystyle u(16u^4 - 20u^2 + 5)$ $=$ $\displaystyle 0$ Factoring a $u$ out of the polynomial $\displaystyle 16u^4 - 20u^2 + 5$ $=$ $\displaystyle 0$ Dividing both sides by $u$

We were able to divide off the $u$ from both sides of that last line because we know that $\cos \dfrac \pi {10}$ cannot possibly equal $0$. From here, we use the quadratic formula to solve for $u^2$.

 $\displaystyle u^2$ $=$ $\displaystyle \dfrac { - \paren { -20 } ± \sqrt { \paren { -20 }^2 - 4 \paren { 16 } \paren 5} } { 2 \paren { 16 } }$ $\displaystyle u^2$ $=$ $\displaystyle \dfrac { 20 ± \sqrt { 80 } } {32}$ $\displaystyle u^2$ $=$ $\displaystyle \dfrac { 5 ± \sqrt 5 } 8$

Taking the square root of both sides, we can solve for $u$.

 $\displaystyle u^2$ $=$ $\displaystyle \dfrac { 5 ± \sqrt 5 } 8$ $\displaystyle u$ $=$ $\displaystyle ± \dfrac 1 {\sqrt 8} \sqrt {5 ± \sqrt 5}$

We know that $\dfrac \pi {10}$ is in the first quadrant, so we can change the exterior $±$ sign to a $+$ sign:

 $\displaystyle u$ $=$ $\displaystyle \dfrac 1 {\sqrt 8} \sqrt {5 ± \sqrt 5}$

We know that $\dfrac \pi {10}$ is a lot closer to $0$ than to $\dfrac \pi 2$:

$\dfrac \pi {10} - 0 < \dfrac \pi 2 - \dfrac \pi {10}$

Therefore, the larger of the possible interior quantities of the radical should be correct.

 $\displaystyle u$ $=$ $\displaystyle \dfrac 1 {\sqrt 8} \sqrt {5 + \sqrt 5}$ $\displaystyle u$ $=$ $\displaystyle \dfrac 1 4 \sqrt { 2 \paren { 5 + \sqrt 5 } }$

Now that we have evaluated $\cos \dfrac \pi {10}$, we can substitute it for the most simplified expression from above with only $\cos \dfrac \pi {10}$:

 $\displaystyle 4 \sqrt { 1 - \cos^2 \dfrac \pi {10} } \paren { 2 \paren { \cos \dfrac \pi {10} }^2 - 1}$ $=$ $\displaystyle 1$ Derived from original theorem $\displaystyle 4 \sqrt { 1 - \paren { \dfrac 1 4 \sqrt { 2 \paren { 5 + \sqrt 5 } } }^2 } \paren { 2 \paren { \dfrac 1 4 \sqrt { 2 \paren { 5 + \sqrt 5 } } }^2 - 1}$ $=$ $\displaystyle 1$ $u = \cos \dfrac \pi {10} = \dfrac 1 4 \sqrt { 2 \paren { 5 + \sqrt 5 } }$ $\displaystyle 4 \sqrt { 1 - \paren { \dfrac 1 4 \sqrt { 2 \paren { 5 + \sqrt 5 } } }^2 } \paren { \dfrac 1 4 \paren { 5 + \sqrt 5 } - 1}$ $=$ $\displaystyle 1$ $2 \paren { \dfrac 1 4 \sqrt { 2 \paren { a } } }^2 = \dfrac 4 {16} a = \dfrac 1 4 a$ $\displaystyle 4 \sqrt { 1 - \dfrac 1 8 \paren { 5 + \sqrt 5 } } \paren { \dfrac 1 4 \paren { 5 + \sqrt 5 } - 1}$ $=$ $\displaystyle 1$ $\paren { \dfrac 1 4 \sqrt { 2 \paren { a } } }^2 = \dfrac 2 {16} a = \dfrac 1 8 a$ $\displaystyle 4 \sqrt { 1 - \dfrac 1 8 \paren { 5 + \sqrt 5 } } \paren { \dfrac { 5 + \sqrt 5 } 4 - 1}$ $=$ $\displaystyle 1$ Multiplication $\displaystyle 4 \sqrt { 1 - \dfrac 1 8 \paren { 5 + \sqrt 5 } } \paren { \dfrac { 5 + \sqrt 5 } 4 - \dfrac 4 4}$ $=$ $\displaystyle 1$ Common denominators $\displaystyle 4 \sqrt { 1 - \dfrac 1 8 \paren { 5 + \sqrt 5 } } \paren { \dfrac { 5 + \sqrt 5 - 4 } 4}$ $=$ $\displaystyle 1$ Combining fractions $\displaystyle 4 \sqrt { 1 - \dfrac 1 8 \paren { 5 + \sqrt 5 } } \paren { \dfrac { 1 + \sqrt 5 } 4}$ $=$ $\displaystyle 1$ Trivial simplification $\displaystyle \dfrac 4 4 \sqrt { 1 - \dfrac 1 8 \paren { 5 + \sqrt 5 } } \paren { 1 + \sqrt 5 }$ $=$ $\displaystyle 1$ Associative property $\displaystyle \sqrt { 1 - \dfrac 1 8 \paren { 5 + \sqrt 5 } } \paren { 1 + \sqrt 5 }$ $=$ $\displaystyle 1$ Trivial simplification $\displaystyle \sqrt { \dfrac 8 8 - \dfrac { 5 + \sqrt 5 } 8 } \paren { 1 + \sqrt 5 }$ $=$ $\displaystyle 1$ Common denominators $\displaystyle \sqrt { \dfrac { 8 - 5 - \sqrt 5 } 8 } \paren { 1 + \sqrt 5 }$ $=$ $\displaystyle 1$ Combining fractions $\displaystyle \sqrt { \dfrac { 3 - \sqrt 5 } 8 } \paren { 1 + \sqrt 5 }$ $=$ $\displaystyle 1$ Trivial simplification $\displaystyle \sqrt { \dfrac { 6 - 2 \sqrt 5 } {16} } \paren { 1 + \sqrt 5 }$ $=$ $\displaystyle 1$ Multiplying the radicand by $\dfrac 2 2$ $\displaystyle \sqrt { \dfrac { 1 - 2 \sqrt 5 + 5 } {16} } \paren { 1 + \sqrt 5 }$ $=$ $\displaystyle 1$ Rewriting $k ± 2 \sqrt u$ as $\paren { k - u } ± 2 \sqrt u + u$ because $k - u + u = k$ $\displaystyle \sqrt { \dfrac { \paren { 1 - \sqrt 5 }^2 } {16} } \paren { 1 + \sqrt 5 }$ $=$ $\displaystyle 1$ Rewriting the quantity as a square $\displaystyle ± \dfrac 1 4 \paren { 1 - \sqrt 5 } \paren { 1 + \sqrt 5 }$ $=$ $\displaystyle 1$ Square root of the radicand $\displaystyle ± \dfrac 1 4 \paren { 1 - 5 }$ $=$ $\displaystyle 1$ Multiplying $\paren { 1 - \sqrt 5 }$ and $\paren { 1 + \sqrt 5 }$ $\displaystyle ± \dfrac 1 4 × -4$ $=$ $\displaystyle 1$ Trivial simplification $\displaystyle ∓ 1$ $=$ $\displaystyle 1$ Multiplication $\displaystyle 1$ $=$ $\displaystyle 1$ Sign rectification

$\blacksquare$