Using the trigonometric identities I have to prove that:
$$ \sin(20^\circ)\cos(65^\circ)-\cos(20^\circ)\sin(65^\circ)=-\frac{1}{\sqrt{2}} $$
I solved
$$ \sin(20^\circ)\cos(65^\circ) = \frac{\sin(-45^\circ)+\sin(85^\circ)}{2}$$
and
$$ -\cos(20^\circ)\sin(65^\circ)=\frac{\sin(45^\circ)+\sin(85^\circ)}{2} $$
How do I continue from here to prove that it is equal $-\dfrac{1}{\sqrt{2}}$
$\endgroup$23 Answers
$\begingroup$Hint: Use the sine of differences:
$$\sin(a-b) = \sin(a)\cos(b) - \cos(a)\sin(b)$$
Using this you can show that the expression equals $\sin(-45^{\circ})$ by inspection.
$\endgroup$$\begingroup$Use fact that
$$ \sin(a - b) = \sin a\cos b - \sin b \cos a $$
$$\text{Therefore, }\, -\sin (65 - 20) = -\sin 65 \cos 20 + \sin 20 \cos 65 = -\left(\frac{1}{\sqrt{2}}\right) $$
$$\text{That is },\, \sin 45 = \frac{1}{\sqrt{2}} $$
$\endgroup$1$\begingroup$This answer seems to match the OP's original intent.
Using the product to sum formula: $$ \sin(a)\cos(b)=\frac{1}{2}(\sin(a+b)+\sin(a-b)), $$ it follows that \begin{align*} \sin(20)\cos(65)&=\frac{1}{2}(\sin(85)+\sin(-45))\\ \sin(65)\cos(20)&=\frac{1}{2}(\sin(85)+\sin(45)). \end{align*}
Then, $$ \sin(20)\cos(65)-\sin(65)\cos(20)=\frac{1}{2}(\sin(85)+\sin(-45))-\frac{1}{2}(\sin(85)+\sin(45)). $$ Since $\sin$ is an odd function, $\sin(-45)=-\sin(45)$. Therefore, the $\sin(85)$'s cancel and you're left with $$ \sin(20)\cos(65)-\sin(65)\cos(20)=-\sin(45). $$ By evaluating this directly, you have the value you're looking for.
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