Using the fact that sin (A+B)=sinAcosB+cosAsinB and the differentiation, obtain the sum formula for cosines.
Solution and Explanation
sin(A+B)=sinAcosB+cosAsinB
Differentiating both sides with respect to x, we obtain
\(\frac{d}{dx}\)[sin(A+B)]=\(\frac{d}{dx}\)(sinAcosB)+\(\frac{d}{dx}\)(cosAsinB)
⇒cos(A+B).\(\frac{d}{dx}\)A+B)=cosB.\(\frac{d}{dx}\)(sinA)+sinA.\(\frac{d}{dx}\)cosB)+sinB.\(\frac{d}{dx}\)(cosA)+cosA.\(\frac{d}{dx}\)(sinB)
⇒cos(A+B).\(\frac{d}{dx}\)(A+B)=cosB.cosA\(\frac{dA}{dx}\)+sinA(-sinB)\(\frac{dB}{dx}\)+sinB(-sinA).\(\frac{dA}{dx}\)+cosAcosB\(\frac{dB}{dx}\)
⇒cos(A+B).[\(\frac{dA}{dx}\)+\(\frac{dB}{dx}\)]=(cosAcosb-sinAsinB).[\(\frac{dA}{dx}\)+\(\frac{dB}{dx}\)]
∴cos(A+B)=cosAcosB-sinAsinB
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Concepts Used:
Continuity & Differentiability
Definition of Differentiability
f(x) is said to be differentiable at the point x = a, if the derivative f ‘(a) be at every point in its domain. It is given by
Definition of Continuity
Mathematically, a function is said to be continuous at a point x = a, if
It is implicit that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at x=a exist and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is unspecified or does not exist, then we say that the function is discontinuous.