`tan^2(2x)cos^4(2x)` Use the power reducing formulas to rewrite the expression in terms of the first power of the cosine.

Friday, February 3, 2012

$\displaystyle{{\tan}}^{{2}}{\left({2}{x}\right)}{{\cos}}^{{4}}{\left({2}{x}\right)}$ Use the power reducing formulas to rewrite the expression in terms of the first power of the cosine.

It is known that

$\displaystyle{{\cos}}^{{2}}{\left({y}\right)}={\left(\frac{{1}}{{2}}\right)}{\left({1}+{\cos{{\left({2}{y}\right)}}}\right)}$ and

$\displaystyle{{\tan}}^{{2}}{\left({y}\right)}=\frac{{{1}-{\cos{{\left({2}{y}\right)}}}}}{{{1}+{\cos{{\left({2}{y}\right)}}}}}.$

Repeating the first formula we obtain

$\displaystyle{{\cos}}^{{4}}{\left({y}\right)}={\left(\frac{{1}}{{4}}\right)}{\left({1}+{2}{\cos{{\left({2}{y}\right)}}}+{{\cos}}^{{2}}{\left({2}{y}\right)}\right)}=$

$\displaystyle={\left(\frac{{1}}{{4}}\right)}{\left({1}+{2}{\cos{{\left({2}{y}\right)}}}+{\left(\frac{{1}}{{2}}\right)}{\left({1}+{\cos{{\left({4}{y}\right)}}}\right)}=\right.}$

$\displaystyle={\left(\frac{{1}}{{8}}\right)}{\left({3}+{4}{\cos{{\left({2}{y}\right)}}}+{\cos{{\left({4}{y}\right)}}}\right)}.$

Finally, for $\displaystyle{y}={2}{x}$

$\displaystyle{{\tan}}^{{2}}{\left({2}{x}\right)}\cdot{{\cos}}^{{4}}{\left({2}{x}\right)}=\frac{{{1}-{\cos{{\left({4}{x}\right)}}}}}{{{1}+{\cos{{\left({4}{x}\right)}}}}}\cdot{\left(\frac{{1}}{{8}}\right)}{\left({3}+{4}{\cos{{\left({4}{x}\right)}}}+{\cos{{\left({8}{x}\right)}}}\right)}.$

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Friday, February 3, 2012

$\displaystyle{{\tan}}^{{2}}{\left({2}{x}\right)}{{\cos}}^{{4}}{\left({2}{x}\right)}$ Use the power reducing formulas to rewrite the expression in terms of the first power of the cosine.

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