Question

Find the value of cos^2 15 - cos^2 75

Answer 1

$\textbf{Given:}$

$\bf\;cos^215-cos^275$

$\text{Using,}\;\bf\;a^2-b^2=(a+b)(a-b)$

$=(cos15-cos75)(cos15+cos75)$

$\text{Using,}$

$\boxed{\bf\;cosC+cosD=2\;cos(\frac{C+D}{2})\;cos(\frac{C-D}{2})}$

$\boxed{\bf\;sinC+sinD=2\:sin(\frac{C+D}{2})\:cos(\frac{C-D}{2})}$

$=(-2\;sin(\frac{15+75}{2})\;sin(\frac{15-75}{2}))(2\;cos(\frac{15+75}{2})\;cos(\frac{15-75}{2}))$

$=(-2\;sin45\;sin(-30))(2\;cos45\;cos(-30))$

$=(2\;sin45\;sin30)(2\;cos45\;cos30)$

$=(2(\frac{1}{\sqrt2})(\frac{1}{2}))(2(\frac{1}{\sqrt2})(\frac{\sqrt3}{2}))$

$=(\frac{1}{\sqrt2})((\frac{1}{\sqrt2})(\sqrt3))$

$=\frac{\sqrt3}{2}$

$\therefore\bf\;cos^215-cos^275=\frac{\sqrt3}{2}$