If $\cos 25^\circ + \sin 25^\circ = k,$ then what is $\cos 20^\circ$?

Question

Question:

If $$\cos 25^\circ + \sin 25^\circ = k,$$ then what is $\cos 20^\circ$?

---

What I did:

I tried to square both sides, and obtained that $\sin 50 = k^2 -1$, however, this didn't get me anywhere. Then I tried splitting 25 into 20 + 5 but that didn't get me anywhere either. Can someone just point me in the right direction?

Answer 1

Hint

Take into account that $20 = 45 -25$. Develop $\displaystyle\cos(45^\circ-25^\circ)$ and remember that $\displaystyle\cos(45^\circ)=\sin(45^\circ)=\frac{\sqrt 2}2$ and see what happens.

I am sure that you can take from here.

Answer 2

Then we have $$k^2=(\cos 25^\circ + \sin 25^\circ)^2=1+2\cos 25^\circ \sin 25^\circ=1+\sin 50^\circ.$$ Note that $\cos 40^\circ=\cos(90^\circ-50^\circ)=\sin 50^\circ$. Hence, we have $$\tag{1}\cos 40^\circ=k^2-1.$$ Now $$\tag{2}\cos 40^\circ=2\cos^2 20^\circ-1.$$ Now combining $(1)$ and $(2)$ gives you the answer.

Answer 3

$cos 25^\circ+ sin 25^\circ=k$

$\implies cos (45-20) + sin (45-20) =k$

$\implies cos 45\cdot cos 20+ sin 45\cdot sin 20 + sin45\cdot cos 20 -cos 45\cdot sin 20=k$

$2\cdot(\frac{1}{\sqrt2})cos 20^\circ=k$

$cos 20^\circ=k\cdot(\frac{\sqrt{2}}{2})$