Question 609036
{{{cos^2(pi/8)-sin^2(pi/8)}}}
First of all, when you see/hear "exact value" in a Trig problem, you should know to put your calculator away. The problem can and must be solved using special angles.<br>
But {{{pi/8}}} is NOT one of our special angles!? So how are we supposed to figure this out? Well, somehow we have to find a way to change each {{{pi/8}}} into one of the special angles. And there are a number of Trig properties/identities that allow you to change the argument of a trig function:<ul><li>Angle sum properties: sin(A+B), cos(A+B), tan(A+B)</li><li>Angle difference properties: sin(A-B), cos(A-B), tan(A-B)</li><li>The double angle properties: sin(2x), cos(2x) [of which there are 3 varieties), tan(2x)</li><li>The half angle properties: {{{sin((1/2)x)}}}, {{{cos((1/2)x)}}} and {{{tan((1/2)x)}}}</li></ul>So which one(s) will help us change {{{pi/8}}} into a special angle. There are two clues:<ul><li>If we notice that {{{2*(pi/8) = pi/4}}} then a double angle property is suggested.</li><li>Your expression fits the pattern of the right side of {{{cos(2x) = cos^2(x) - sin^2(x)}}}.
NOTE: Trig will be a lot easier if you learn that these properties are patterns. The x's, A's and B's in all these properties are just placeholders. They can be replaced by <i>any mathematical expression</i> and the equation will still be true!! For example:
{{{cos(2y) = cos^2(y) - sin^2(y)}}}
{{{cos(2(1.7)) = cos^2(1.7) - sin^2(1.7)}}}
{{{cos(2(3x+4)) = cos^2(3x+4) - sin^2(3x+4)}}}
{{{cos(2(x^2-4x+10)) = cos^2(x^2-4x+10) - sin^2(x^2-4x+10)}}}
etc.</li></ul>
So we will use the cos(2x) property to rewrite
{{{cos^2(pi/8)-sin^2(pi/8)}}}
as
{{{cos(2*(pi/8))}}}
which simplifies as follows:
{{{cos(pi/4)}}}
{{{sqrt(2)/2}}}