<s>[INST] Find the value of $\cos105^\circ+\sin75^\circ$ Find the value of
$$\cos105^\circ+\sin75^\circ.$$
We can write the given trig expression as $$\cos(180^\circ-75^\circ)+\sin75^\circ=-\cos75^\circ+\sin75^\circ\\=\sin75^\circ-\cos75^\circ$$
I don't see what else I can do. Thank you!
 [/INST] There's some transformation formula I'm listing below,
$\sin (a+b) +  \sin (a-b)= 2 \sin a \cos b$
$\sin (a+b)-\sin (a-b) = 2\cos a \sin b$
$\cos (a+b)+ \cos (a-b) = 2\cos a \cos b$
$\cos (a-b) -\cos (a+b) = 2\sin a \sin b$
Now your problem reduces to $\sin 75° - \cos 75°$, to make use the above formula, we convert $\cos 75°= \sin 15°$ and we write 75 = 45+30 and 15 = 45-30.
So we have $\sin (45°+30°) - \sin (45°-30°) = 2\cos 45° \cdot \sin 30° = 2\cdot \dfrac{1}{√2}\cdot \dfrac{1}{2}= \dfrac{1}{\sqrt{2}}$
Alter: You need not break the given form even, as follows
$\cos 105° + \sin 75°= \cos 105° +\cos 15°= \cos(60+45)+\cos (60-45)$
Check from the above given formulas, and you should get the same answer.
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