Question 1205464
<br>
The equation you show is NOT an identity.<br>
You can see that quickly by trying x=0, where cos(x)=1 and sin(x)=0.  The equation then says<br>
{{{0+1+0(1)+0(1)=0(1)}}}
{{{0+1+0+0=0}}}<br>
You apparently have not shown the right side of the equation correctly.<br>
{{{(sin(x))^3+(cos(x)^3)+(sin(x)(cos(x)^2))+(sin(x)^2)(cos(x))}}}<br>
Group the terms in pairs...<br>
{{{(sin(x)^3)+(sin(x)^2)(cos(x))+(cos(x)^3)+(cos(x)^2)(sin(x))}}}<br>
...factor out the common factor in each pair...<br>
{{{(sin(x)^2)(sin(x)+cos(x))+(cos(x)^2)(sin(x)+cos(x))}}}<br>
...and find the common factor (sinx+cosx)...<br>
{{{(sin(x)^2+cos(x)^2)(sin(x)+cos(x))}}}<br>
...and then use sin^2x+cos^2x=1:<br>
{{{(1)(sin(x)+cos(x))=sin(x)+cos(x)}}}<br>
The expression on the left is equivalent to sinx+cosx -- not to (sinx)(cosx).<br>