Geometry Level 4

In a triangle $$\Delta XYZ$$, let $$a,b$$ and $$c$$ be the lengths of the sides opposite to the angles $$X,Y$$ and $$Z$$, respectively. If $$2(a^2-b^2)=c^2$$ and $$\lambda=\dfrac{\sin(X-Y)}{\sin Z}$$, then find the possible value(s) of $$n$$ for which $$\cos(n\pi\lambda)=0$$. $\begin{array}{} (1) \, 5 \quad \quad \quad \quad \quad \quad \quad \quad & (2) \, 3 \\ (3) \, 2 & (4) \, 1 \end{array}$ Note :

• Submit your answer as the increasing order of the serial numbers of all the correct options.

• For eg, if your answer is $$(1),(2)$$, then submit 12 as the correct answer, if your answer is $$(2),(3),(4)$$, then submit 234 as the correct answer.

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