Question 1152225
<pre>
Let <font face="symbol">a</font> = x + x²
Let <font face="symbol">b</font> = x - x²

{{{matrix(1,3,abs(A),""="",

abs(matrix(3,3,

cos(alpha^""),   sin(alpha^""),  -cos(alpha^""),
sin(beta^""),    cos(beta^""),   sin(beta^""),  
 sin(2x^""),           0,      sin(2x^2) )))}}} 

Expand about the bottom row:

{{{sin(2x^"")(sin(alpha^"")sin(beta^"")+cos(alpha^"")^""cos(beta^""))+
sin(2x^2)(cos(alpha^"")cos(beta^"")-sin(alpha^"")^""sin(beta^""))}}}

Swap the terms in the first big parentheses so you'll recognize the identity:

{{{sin(2x^"")(cos(alpha^"")cos(beta^"")+sin(alpha^"")^""sin(beta^""))+
sin(2x^2)(cos(alpha^"")cos(beta^"")-sin(alpha^"")^""sin(beta^""))}}}

Using identities for cos(<font face="symbol">a</font> ± <font face="symbol">b</font>)

{{{sin(2x^"")(cos(alpha-beta)^"")+
sin(2x^2)(cos(alpha+beta)^"")}}}

Swapping the factors in the second term so you'll recognize the identity:

{{{sin(2x^"")cos(alpha-beta)^""+
cos(alpha+beta)sin(2x^2)^""}}}

Since <font face="symbol">a</font> = x + x² and<font face="symbol">b</font> = x - x², <font face="symbol">a</font>+<font face="symbol">b</font> = 2x and <font face="symbol">a</font>-<font face="symbol">b</font> = 2x² 

{{{sin(2x^"")cos(2x^2)^""+
cos(2x)sin(2x^2)^""}}}

Using identity for sin(<font face="symbol">a</font> + <font face="symbol">b</font>),

{{{sin(2x+2x^2)}}}

Edwin</pre>