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Let y = cos2x sin3x then \left(\frac{d y^{105}}{d x^{105}}\right)_{x=0}=

Option: 1

\frac{1}{16}\left[2+3^{105}-5^{105}\right]


Option: 2

\frac{1}{8}\left[2+105^3-105^5\right]


Option: 3

\frac{1}{16}\left[3+2^{105}-105^5\right]


Option: 4

\frac{1}{8}\left[2-105^3+105^5\right]


Answers (1)

best_answer

Lety=\cos ^2 x \sin ^3 x=\frac{1}{2}(1+\cos 2 x) \frac{1}{4}(3 \sin x-\sin 3 x)\begin{aligned} & =\frac{1}{8}(3 \sin x-\sin 3 x+3 \sin x \cos 2 x-\sin 3 x \cos 2 x) \\ & =\frac{1}{16}(2 \sin x+\sin 3 x-\sin 5 x) \end{aligned}

now using the standard formula Dnsin(ax+(B), we get

y_n=\frac{1}{16}\left[2 \sin \left(x+\frac{1}{2} n \pi\right)+3^n \sin \left(3 x+\frac{1}{2} n \pi\right)-5^2 \sin \left(5 x+\frac{1}{2} n \pi\right)\right]

Posted by

himanshu.meshram

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