Question: Problems (a.) through (e.) indicate how properties of \(\sin (x)\) and \(\cos (x)\) can be obtained directly from the fact that these functions are solutions of certain initial value problems. Imagine for the moment that \(\sin (x)\) and \(\cos (x)\) are unfamiliar. Let \(S(x)\) be the unique solution of the initial value problem

Let \(C(x)\) be the unique solution of

1. Show that \(S^{\prime}(x)=C(x)\).
2. Show that \(C^{\prime}(x)=-S(x)\).
3. Show that \(S^{2}(x)+C^{2}(x)=1\) for all $x .$ Hint: Consider the derivative of \(\varphi(x)=S^{2}(x)+C^{2}(x)\)
4. Show that \(C\) and \(S\) are linearly independent solutions of \(y^{\prime \prime}+y=0\).
5. Show that \(S(a+b)=S(a) C(b)+C(a) S(b)\) for every pair of real numbers \(a\) and \(b\). Hint: Consider the initial value problem

Show that \(\varphi(x)=c_{1} C(x)+c_{2} S(x)\) is the general solution of the differential equation, then determine \(c_{1}\) and \(c_{2}\) from the original data. Next, show that \(\psi(x)=S(x+a)\) is also a solution of this initial value problem.

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Deliverable: Word Document