Question

Вычислите.

cos105-sin 195+sin(-135)-cos135

tg18*tg288+sin32*sin148-sin302*sin122

 

Answer 1

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

$mathrm{tg}18^circmathrm{tg}288^circ+sin32^circsin148^circ-sin302^circsin122^circ= \ =mathrm{tg}18^circmathrm{tg}(270^circ+18^circ)+sin32^circsin(180^circ-32^circ)- \ -sin(360^circ-58^circ)sin(180^circ-58^circ)= \ =mathrm{tg}18^circ(-mathrm{ctg}18^circ)+sin32^circsin32^circ -(-sin58^circ)sin58^circ=$
$=-mathrm{tg}18^circmathrm{ctg}18^circ+sin^232^circ+sin^258^circ= \ =-1+sin^232^circ+sin^2(90^circ-32^circ)= \ =-1+sin^232^circ+cos^232^circ=-1+1=0$