how to prove this question of eigen-values and eigen-vectors?

Question

If $$ A=\begin{bmatrix} \sin\theta & \csc\theta & 1 \\ \sec\theta & \cos\theta & 1 \\ \tan\theta & \cot\theta & 1 \\ \end{bmatrix} $$ then prove that there does not exist a real value of $\theta$ for which characteristics roots of $A$ are $-1,1,3$

i tried to solve as follows, sum of eigen value $$=\sin\theta +\cos\theta + 1=-1+1+3=3$$ $$\sin\theta +\cos\theta = 2$$ but what to do next.

Answer 1

\begin{eqnarray} a\sin\theta+b\cos\theta&=&\sqrt{a^2+b^2}\left[\frac{a}{\sqrt{a^2+b^2}}\sin\theta+\frac{b}{\sqrt{a^2+b^2}}\cos\theta\right]\\ &=&\sqrt{a^2+b^2}\left[\sin\phi\sin\theta+\cos\phi\cos\theta\right]\\ &=&\sqrt{a^2+b^2}\cos(\theta-\phi) \end{eqnarray}

Where $$\phi=\arcsin\frac{a}{\sqrt{a^2+b^2}}$$

So the result is a sinusoidal with a phase shift and an amplitude of $\sqrt{a^2+b^2}$

When $a=b=1$ what will be the amplitude?

Answer 2

$$\cos x+\sin x=2\implies(\cos x+\sin x)^2=4\implies\cos^2+2\cos x\sin x+\sin^2x=4\\\implies\sin2x=3\ (!)$$