$$ \begin{align} x_1 &= A \; cos \;\omega t \\[4pt] x_2 &= A \; cos \; \omega t \\[4pt] x &= x_1 + x_2 = 2A \; cos \; \omega t \end{align} $$
$$ \begin{align} x_1 &= A \; cos \; \omega t \\[4pt] x_2 &= A \; cos \; (\omega t + \pi) \\[4pt] x &= x_1+x_2 = 0 \end{align} $$
$$ \begin{align} x_1 &= A \; cos \; \omega t \\[4pt] x_2 &= A \; cos \; (\omega t + \pi/2) \\[4pt] x &= x_1+x_2 = \sqrt 2 A \; cos \; (\omega t + \pi/4) \end{align} $$
$$ \begin{align} x_1 &= A \; cos \; \omega t \\[4pt] x_2 &= A \; cos \; \omega t + 2\pi/3 \\[4pt] x &= x_1+x_2 = 2A \; cos \; \omega t + \pi/3 \end{align} $$
$$ x_1 = A \; cos \; \omega_1 t \; \text {and} \; x_2 = A \; cos \;\omega_2 t $$ $$ x = x_1+x_2 = A(t) \; cos \; \omega t \; \text{where} \; $$ $$ A(t) = 2A \; cos \; \left\{ \frac {\omega_1 - \omega_2} {2} \right\}\; \text {and}\; \omega = \left \{ \frac{\omega_1+\omega_2}{2} \right\} $$
$$ \begin{align} x_1 &= A \; cos \;\omega t \\[4pt] x_2 &= A \; cos \; \omega t \\[4pt] x &= x_1 + x_2 = 2A \; cos \; \omega t \end{align} $$
$$ \begin{align} x_1 &= A \; cos \; \omega t \\[4pt] x_2 &= A \; cos \; (\omega t + \pi) \\[4pt] x &= x_1+x_2 = 0 \end{align} $$
$$ \begin{align} x_1 &= A \; cos \; \omega t \\[4pt] x_2 &= A \; cos \; (\omega t + \pi/2) \\[4pt] x &= x_1+x_2 = \sqrt 2 A \; cos \; (\omega t + \pi/4) \end{align} $$
$$ \begin{align} x_1 &= A \; cos \; \omega t \\[4pt] x_2 &= A \; cos \; \omega t + 2\pi/3 \\[4pt] x &= x_1+x_2 = 2A \; cos \; \omega t + \pi/3 \end{align} $$
$$ x_1 = A \; cos \; \omega_1 t \; \text {and} \; x_2 = A \; cos \;\omega_2 t $$ $$ x = x_1+x_2 = A(t) \; cos \; \omega t \; \text{where} \; $$ $$ A(t) = 2A \; cos \; \left\{ \frac {\omega_1 - \omega_2} {2} \right\}\; \text {and}\; \omega = \left \{ \frac{\omega_1+\omega_2}{2} \right\} $$
