Question

${\cos }^2{5}^{\circ }-{\cos }^{2}15^{\circ }-{\sin }^{2}15^{\circ }+{\sin }^{2}35^{\circ }$ $+\cos 15^{\circ }\sin 15^{\circ }-\cos 5^{\circ }\sin 35^{\circ }=$

• A. 0
• B. 1
• C. $\frac{3}{2}$
• D. 2

Answer 1

Answer: (A)

${\cos }^2{5}^{\circ }-{\cos }^{2}15^{\circ }-{\sin }^{2}15^{\circ }+{\sin }^{2}35^{\circ }$
$+\cos 15^{\circ }\sin 15^{\circ }-\cos 5^{\circ }\sin 35^{\circ }$
$=\cos 5^{\circ }\left(\cos 5^{\circ }-\sin 35^{\circ }\right)-\cos 15^{\circ }\left(\cos 15^{\circ }-\sin 15^{\circ }\right)$
$+{\sin }^{2}35^{\circ }-{\sin }^{2}15^{\circ }$
$=\cos 5^{\circ }\left(\cos 5^{\circ }-\cos 55^{\circ }\right)-\cos 15^{\circ }\left(\cos 15^{\circ }-\cos 75^{\circ }\right)$
$+\sin 50^{\circ }\sin 20^{\circ }$
$=\cos 5^{\circ }\left(2\sin 30^{\circ }\sin 25^{\circ }\right)-\cos 15^{\circ }\left(2\sin 45^{\circ }\sin 30^{\circ }\right)$
$+\sin 50^{\circ }\sin 20^{\circ }$
$=-\frac{\cos 15^{\circ }}{\sqrt{2}}+\cos 5^{\circ }\sin 25^{\circ }+\sin 50^{\circ }\sin 20^{\circ }$
$=-\frac{\cos 15^{\circ }}{\sqrt{2}}+\frac{1}{2}\left(2\cos 5^{\circ }\cos 65^{\circ }\right)+\frac{1}{2}\left(2\sin 50^{\circ }\sin 20^{\circ }\right)$
$=-\frac{\cos 15^{\circ }}{\sqrt{2}}+\frac{1}{2}\left[\cos 70^{\circ }+\cos 60^{\circ }+\cos 30^{\circ }-\cos 70^{\circ }\right]$
$=-\frac{\cos 5^{\circ }}{\sqrt{2}}+\frac{1}{2}\left[2\cos 45^{\circ }\cos 15^{\circ }\right]$
$=-\frac{\cos 15^{\circ }}{\sqrt{2}}+\frac{\cos 15^{\circ }}{\sqrt{2}}=0$