Question

$\huge\mathfrak{Prove:}$

$\bf{\frac{d( \ { \sin}^{n}x . \cos \: n \: x ) }{dn} = n \: { \sin }^{n + 1} \: \cos( n + 1)x}$


$\huge \mathcal \blue{Don't Spam}$​

Answer 1

${\large{\underbrace{\mathbb{\pink{ ANSWER\: \:-: }}}}}$

If n is a positive integer, prove that

ddx(sinnxcosnx)

=nsinn−1xcos(n+1)

x.Proof of Derivative Value:

If we have a trig-function with exponent m such as cosm(x),

then it is also written as the trig-function with a negative exponent using exponent rule.

1xm

=x−m

⇒cosm(x)

=1cos−m(x),

where,

m is the positive integer.

To proof the value of the derivative of trig-functions, we'll apply the sum rule of the cosine function in the solution of derivative to get the exact function on the right-hand side of the given derivative value.

cos(A+B)

=cosAcosB−sin Asin B

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━☞Final Answer :-

cosAcosB−sin Asin B

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