Is the function $A\cos(\lambda x) + B\sin(\lambda x)$ periodical? I am trying to check the function for periodicity... $y(x) = A\cos(\lambda x) + B\sin(\lambda x) $ I think that it's possible to rewrite the function as $y(x) = Csin (\lambda x + t)$, where $C = \sqrt{A^2 + B^2}$ But I can't prove that new function is periodical or is not periodical. How can I do it?

**Hint:**

The period of the sine and cosine functions are well known to be $2\pi$ for both. Hence $\dfrac{2\pi}\lambda$ is a period of the linear combination, for the argument $\lambda x$.

Remains to show that it is the smallest.

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Setting $t:=\lambda x$, let $T=\lambda X$ be the period.

$$A\cos(t+T)+B\sin(t+T)=A\cos(t)+B\sin(t)$$

implies, using the sum-to-product formula,

$$-2A\sin\left(t+\frac T2\right)\sin\left(\frac T2\right)+2A\cos\left(t+\frac T2\right)\sin\left(\frac T2\right)=0.$$

This expression is identically zero for the smallest nonzero value $T=2\pi$.