Barbara tried to calculate $$\cos465^{\circ}$$ without using a calculator. She proceeded as follows: \begin{aligned} \cos465^{\circ}&\stackrel{(1)}=\cos\left(360^{\circ}+105^{\circ}\right)\stackrel{(2)}=\cr &\stackrel{(2)}=\cos105^{\circ}\stackrel{(3)}=\cos60^{\circ}+\cos45^{\circ}\stackrel{(4)}=\cr &\stackrel{(4)}=\frac12+\frac{\sqrt2}{2}\stackrel{(5)}=\frac{1+\sqrt2}{2} \end{aligned} Did Barbara make a mistake in her solution? If so, specify in which step.
No. Barbara´s solution is correct.
Yes. The mistake is in step (2). The correct simplification should be: $$\cos\left(360^{\circ}+105^{\circ}\right)=1+\cos105^{\circ}$$
Yes. The mistake is in step (3). It should be: $$\cos105^{\circ}=\cos60^{\circ}\cos45^{\circ}-\sin60^{\circ}\sin45^{\circ}$$
Yes. The mistake is in step (4). It should be: $$\cos60^{\circ}+\cos45^{\circ}=\frac{\sqrt3}{2}+\frac{\sqrt2}{2}$$
Barbara made a mistake in step (3). She should have used the trigonometric angle sum identity: $$\cos(x+y)=\cosx\cosy-\sinx\siny$$ Correct solution: \begin{aligned} \cos465^{\circ}&=\cos\left(360^{\circ}+105^{\circ}\right)=\cos105^{\circ}=\cr &=\cos\left(60^{\circ}+45^{\circ}\right)=\cos60^{\circ}\cos45^{\circ}-\sin60^{\circ}\sin45^{\circ}=\cr &=\frac12\cdot\frac{\sqrt2}{2}-\frac{\sqrt3}{2}\cdot\frac{\sqrt2}{2}=\frac{\sqrt2}{4}\left(1-\sqrt3\right) \end{aligned}