Question

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Answer 1

$1)\ \ \dfrac{sin40-cos40}{\sqrt2\cdot sin15}=\dfrac{2sin20\cdot cos20-(2cos^220-1)}{\sqrt2\cdot sin15}=\dfrac{2cos20\xsot (sin20-cos20)+1}{\sqrt2\cdot \frac{\sqrt3-1}{2\sqrt2}}=\\\\\\=\dfrac{4cos20(sin20-cos20)+1}{\sqrt3-1}$

$2)\ \ \dfrac{\sqrt2(cos25-cos65)}{sin20}=\dfrac{\sqrt2(2sin45\cdot sin20)}{sin20}=\sqrt2\cdot 2\cdot\dfrac{\sqrt2}{2}=2$

$3)\ \ sina=-\dfrac{1}{2}\\\\\pi$

$4)\ \ cos^241+cos79\cdot cos19-1=(cos^241-1)+\dfrac{1}{2}\cdot (cos98+cos60)=\\\\=-sin^241+\dfrac{cos98}{2}+\dfrac{1}{4}$

$5)\ \ sin^216+cos46\, cos14+1=sin^216+\dfrac{1}{2}\cdot (cos60+cos30)+1=\\\\=sin^216+\dfrac{1}{2}\cdot (\dfrac{1}{2}+\dfrac{\sqrt3}{2})+1=sin^216+\dfrac{1}{2}\cdot \dfrac{1+\sqrt3}{2}+1=sin^216+\dfrac{5+\sqrt3}{4}$