Continuity and Differentiability — Maths STD 12 Science — Question

Consider, A and B as function of t and differentiating both sides w.r.t. x,

$\cos \left( {A + B} \right)\left( {\frac{{dA}}{{dt}} + \frac{{dB}}{{dt}}} \right) = \sin A\left( { - \sin B} \right)\frac{{dB}}{{dt}} + \cos B\left( {\cos A\frac{{dA}}{{dt}}} \right)$$ + \cos A\cos B\frac{{dB}}{{dt}} + \sin B\left( { - \sin A} \right)\frac{{dA}}{{dt}}$ 

$\Rightarrow \cos \left( {A + B} \right)\left( {\frac{{dA}}{{dB}} + \frac{{dB}}{{dt}}} \right) = \left( {\cos A\cos B - \sin A\sin B} \right)\left( {\frac{{dA}}{{dt}} + \frac{{dB}}{{dt}}} \right)$

$\Rightarrow \cos \left( {A + B} \right) = \left( {\cos A\cos B - \sin A\sin B} \right)$