Question

Вычислите. cos105-sin 195+sin(-135)-cos135 tg18*tg288+sin32*sin148-sin302*sin122

Answer 1

$\cos105^\circ-\sin 195^\circ+\sin(-135^\circ)-\cos135^\circ=$
$=\cos(90^\circ+15^\circ)-\sin (180^\circ+15^\circ)+\sin(-180^\circ+45^\circ)-\cos(180^\circ-45^\circ)=$
$=-\sin15^\circ-(-\sin 15^\circ)-\sin45^\circ-(-\cos45^\circ)= \\\ =-\sin15^\circ+\sin 15^\circ-\sin45^\circ+\cos45^\circ=- \dfrac{ \sqrt{2} }{2} + \dfrac{ \sqrt{2} }{2} =0$

$\mathrm{tg}18^\circ\mathrm{tg}288^\circ+\sin32^\circ\sin148^\circ-\sin302^\circ\sin122^\circ= \\\ =\mathrm{tg}18^\circ\mathrm{tg}(270^\circ+18^\circ)+\sin32^\circ\sin(180^\circ-32^\circ)- \\\ -\sin(360^\circ-58^\circ)\sin(180^\circ-58^\circ)= \\\ =\mathrm{tg}18^\circ(-\mathrm{ctg}18^\circ)+\sin32^\circ\sin32^\circ -(-\sin58^\circ)\sin58^\circ=$
$=-\mathrm{tg}18^\circ\mathrm{ctg}18^\circ+\sin^232^\circ+\sin^258^\circ= \\\ =-1+\sin^232^\circ+\sin^2(90^\circ-32^\circ)= \\\ =-1+\sin^232^\circ+\cos^232^\circ=-1+1=0$