`cos^2(x) - 1/2` Use a double angle formula to rewrite the expression.

Friday, May 2, 2014

$\displaystyle{{\cos}}^{{2}}{\left({x}\right)}-\frac{{1}}{{2}}$ Use a double angle formula to rewrite the expression.

You need to use the double angle formula to re-write the expression, such that:

$\displaystyle{{\cos}}^{{2}}{x}-\frac{{1}}{{2}}=\frac{{{2}{{\cos}}^{{2}}{x}-{1}}}{{2}}=\frac{{{\cos{{2}}}{x}}}{{2}}$

$\displaystyle{\cos{{2}}}{x}={\cos{{\left({x}+{x}\right)}}}={\cos{{x}}}\cdot{\cos{{x}}}-{\sin{{x}}}\cdot{\sin{{x}}}$

$\displaystyle{\cos{{2}}}{x}={{\cos}}^{{2}}{x}-{{\sin}}^{{2}}{x}$

Replacing $\displaystyle{1}-{{\cos}}^{{2}}{x}$ for $\displaystyle{{\sin}}^{{2}}{x}$ yields:

$\displaystyle{\cos{{2}}}{x}={{\cos}}^{{2}}{x}-{1}+{{\cos}}^{{2}}{x}$

$\displaystyle{\cos{{2}}}{x}={2}{{\cos}}^{{2}}{x}-{1}$

Hence, using the double angle formula to re-write the expression yields $\displaystyle{{\cos}}^{{2}}{x}-\frac{{1}}{{2}}=\frac{{{\cos{{2}}}{x}}}{{2}}$

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Friday, May 2, 2014

$\displaystyle{{\cos}}^{{2}}{\left({x}\right)}-\frac{{1}}{{2}}$ Use a double angle formula to rewrite the expression.

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