Question 1144070
<pre>Using DeMoivre's theorem:

{{{(cos^""(theta)+i*sin^""(theta)^"")^7=(1(cos(theta)+i*sin(theta)^""  )^"")^7= 1^7(cos(7theta)^""+i*sin(7theta))=

cos(7theta)^""+i*sin(7theta)}}}

Using the binomial theorem:

{{{(a+b)^7 = a^7 + 7a^6b + 21a^5b^2 + 35a^4b^3 + 35a^3b^4 + 21a^2b^5 + 7ab^6 + b^7}}}

{{{(cos^""(theta)+i*sin^""(theta)^"")^7 = cos^7(theta) + 7cos^6(theta)i*sin^""(theta) + 21cos^5(theta)i^2*sin^2(theta) + 35cos^4(theta)i^3*sin^3(theta) + 35cos^3(theta)i^4*sin^4(theta) + 21cos^2(theta)i^5*sin^5(theta) + 7cos^""(theta)i^6*sin^6(theta) + i^7*sin^7(theta)}}}

Then we use

i<sup>2</sup> = -1
i<sup>3</sup> = -i
i<sup>4</sup> = 1
i<sup>5</sup> = i
i<sup>6</sup> = -1
i<sup>7</sup> = -i

{{{(cos^""(theta)+i*sin^""(theta)^"")^7 = cos^7(theta) + 7cos^6(theta)i*sin^""(theta) + 21cos^5(theta)(-1)*sin^2(theta) + 35cos^4(theta)(-i)*sin^3(theta) + 35cos^3(theta)(1)*sin^4(theta) + 21cos^2(theta)(i)*sin^5(theta) + 7cos^""(theta)(-1)*sin^6(theta) + (-i)*sin^7(theta)}}}

We can equate the two expressions for cos(7<font face="symbol">q</font>) + i∙sin(7<font face="symbol">q</font>)

{{{cos^""(7theta)+i*sin^""(7theta) = cos^7(theta) + 7cos^6(theta)i*sin^""(theta) + 21cos^5(theta)(-1)*sin^2(theta) + 35cos^4(theta)(-i)*sin^3(theta) + 35cos^3(theta)(1)*sin^4(theta) + 21cos^2(theta)(i)*sin^5(theta) + 7cos^""(theta)(-1)*sin^6(theta) + (-i)*sin^7(theta)}}}

We equate the REAL parts on each side:

{{{cos^""(7theta) = cos^7(theta) + 21cos^5(theta)(-1)*sin^2(theta) + 35cos^3(theta)(1)*sin^4(theta) + 7cos(theta)(-1)*sin^6(theta)}}}

{{{cos^""(7theta) = cos^7(theta) - 21cos^5(theta)sin^2(theta) + 35cos^3(theta)sin^4(theta) - 7cos(theta)sin^6(theta)}}}

We equate the IMAGINARY parts on both sides:

{{{i*sin^""(7theta) = 7cos^6(theta)i*sin^""(theta) + 35cos^4(theta)(-i)*sin^3(theta) + 21cos^2(theta)(i)*sin^5(theta) + (-i)*sin^7(theta)}}}

Dividing through by i:

{{{sin^""(7theta) = 7cos^6(theta)sin^""(theta) - 35cos^4(theta)sin^3(theta) + 21cos^2(theta)sin^5(theta) - sin^7(theta)}}}

So we have identities for both cos(7<font face="symbol">q</font>) and sin(7<font face="symbol">q</font>)

You do the other one the same way.  I'll help you with the binomial theorem part:

{{{(a+b)^6=a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6}}}

Edwin</pre>