Notes: `cos((5pi)/4 - x) = -sqrt(2)/2 (cos(x) + sin(x))` Prove the identity.

Wednesday, 18 November 2015

$\displaystyle{\cos{{\left(\frac{{{5}\pi}}{{4}}-{x}\right)}}}=-\frac{\sqrt{{{2}}}}{{2}}{\left({\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}}\right)}$ Prove the identity.

Prove the identity $\displaystyle{\cos{{\left(\frac{{{5}\pi}}{{4}}+{x}\right)}}}=-\frac{\sqrt{{2}}}{{2}}{\left({\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}}\right)}$

Use the formula $\displaystyle{\cos{{\left({u}\right)}}}{\cos{{\left({v}\right)}}}+{\sin{{\left({u}\right)}}}{\sin{{\left({v}\right)}}}$

on the left side of the equation.

$\displaystyle{\cos{{\left(\frac{{{5}\pi}}{{4}}+{x}\right)}}}$

$\displaystyle={\cos{{\left(\frac{{{5}\pi}}{{4}}\right)}}}{\cos{{\left({x}\right)}}}+{\sin{{\left(\frac{{{5}\pi}}{{4}}\right)}}}{\sin{{\left({x}\right)}}}$

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

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

Prove the identity $\displaystyle{\cos{{\left(\frac{{{5}\pi}}{{4}}+{x}\right)}}}=-\frac{\sqrt{{2}}}{{2}}{\left({\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}}\right)}$

Use the formula $\displaystyle{\cos{{\left({u}\right)}}}{\cos{{\left({v}\right)}}}+{\sin{{\left({u}\right)}}}{\sin{{\left({v}\right)}}}$

on the left side of the equation.

$\displaystyle{\cos{{\left(\frac{{{5}\pi}}{{4}}+{x}\right)}}}$

$\displaystyle={\cos{{\left(\frac{{{5}\pi}}{{4}}\right)}}}{\cos{{\left({x}\right)}}}+{\sin{{\left(\frac{{{5}\pi}}{{4}}\right)}}}{\sin{{\left({x}\right)}}}$

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

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

Notes

Wednesday, 18 November 2015

$\displaystyle{\cos{{\left(\frac{{{5}\pi}}{{4}}-{x}\right)}}}=-\frac{\sqrt{{{2}}}}{{2}}{\left({\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}}\right)}$ Prove the identity.

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Is there any personification in "The Tell-Tale Heart"?

Wednesday, 18 November 2015

Prove the identity.

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Post a Comment

Is there any personification in &quot;The Tell-Tale Heart&quot;?

$\displaystyle{\cos{{\left(\frac{{{5}\pi}}{{4}}-{x}\right)}}}=-\frac{\sqrt{{{2}}}}{{2}}{\left({\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}}\right)}$ Prove the identity.

Is there any personification in "The Tell-Tale Heart"?