Value of $\cos 20^\circ + 2 \sin^2 55^\circ - \sqrt{2} \sin 65^\circ$

Question

Value of $\cos 20^\circ + 2 \sin^2 55^\circ - \sqrt{2} \sin 65^\circ$.

Frankly speaking, I have no idea how to start in this question. I tried simplifying $\cos 20^\circ$ in terms of $\cos 55^\circ$ & $35^\circ$ and $\sin 65^\circ$ in terms of $\sin 30^\circ$ & $35^\circ$, however this didn't help.

Any hint would be extremely helpful. :)

Answer 1

First the $2\sin^2$ motivates using double angle: $2\sin^2(55^{\circ})=1-\cos(110^{\circ})=1+\sin(20^{\circ})$.

So we want to find $1+\cos(20^{\circ})+\sin(20^{\circ})-\sqrt{2}\sin(65^{\circ})$.

Now the $65^{\circ}$ is not a nice angle, and the $\sqrt{2}$ isn't too nice either. Let us attempt to square and apply double angle formulas.

We have $(\sqrt{2}\sin(65^{\circ}))^2=2\sin^2(65^{\circ})=1-\cos(130^{\circ})=1+\sin(40^{\circ})$.

$(\cos(20^{\circ})+\sin(20^{\circ}))^2=1+2\sin(20^{\circ})\cos(20^{\circ})=1+\sin(40^{\circ})$.

Note both $\cos(20^{\circ})+\sin(20^{\circ})$ and $\sqrt{2}\sin(65^{\circ})$ are positive, hence they are equal.

Thus we get the final answer as $1$.

P.S. Though, the approach outlined by @Ekaveera Kumar Sharma might be a more systematic way to go about it.

Answer 2

Note first that

$$\sqrt{2}\sin(65) = \sqrt{2}\sin(45+20) = \sqrt{2}\sin(45)\cos(20)+\sqrt{2}\sin(20) \cos(45)\\ = \sin(20)+\cos(20),$$

which leaves you with

$$2\sin^2(55)-\cos(20).$$

Continue using trigonometric identities.

Answer 3

$$\cos 20^{\circ} + 2\sin^2 55^{\circ} - \sqrt{2}\sin 65^{\circ} \\ = \cos 20^{\circ}+ 1 - \cos 110^{\circ} - \sqrt{2}\sin 65^{\circ} \\ = 2\sin(\frac{110^{\circ}+20^{\circ}}{2})\sin(\frac{110^{\circ} - 20^{\circ}}{2}) - \sqrt{2}\sin 65^{\circ} +1\\ = 2\sin 45^{\circ}\sin65^{\circ} - \sqrt{2}\sin 65^{\circ} +1 \\ = \sqrt{2}\sin 65^{\circ} - \sqrt{2}\sin 65^{\circ} +1\\ = 1$$

Answer 4

 cos20∘+ 2sin^2(55)-2√sin65∘
=cos20∘+1−cos110∘−2√sin65∘
=-2sin(110∘+20∘/2)sin(110∘−20∘/2)−2√sin65∘+1 [Because cos(x)-cos(y)=-2{sin(x+y)/2}{sin(x-y)/2}]
=-2sin45∘sin65∘−2√sin65∘+1
=-2√sin65∘−2√sin65∘+1
=-4√sin65∘ + 1