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  • Find the value of experssion <img alt="\tan \alpha+2 \tan 2\alpha A+4 \tan 4 \alpha+8 \tan 8 \alpha+ 16 \cot 16 \alpha" src="https://learn.careers360.com/latex-image/?%5Ctan%20%5Calpha&plu

Find the value of experssion \tan \alpha+2 \tan 2\alpha A+4 \tan 4 \alpha+8 \tan 8 \alpha+ 16 \cot 16 \alpha

Option: 1

\tan \alpha


Option: 2

\tan \alpha+\cot\alpha


Option: 3

\cot \alpha


Option: 4

1


Answers (1)

best_answer

Sum/Difference into Product -

Sum/Difference into Product-

 

\\1.\;\;\sin \alpha+\sin \beta=2 \sin \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)\\\\2.\;\;\sin \alpha-\sin \beta=2 \sin \left(\frac{\alpha-\beta}{2}\right) \cos \left(\frac{\alpha+\beta}{2}\right)\\\\3.\;\;\cos \alpha-\cos \beta=-2 \sin \left(\frac{\alpha+\beta}{2}\right) \sin \left(\frac{\alpha-\beta}{2}\right)\\\\4.\;\;\cos \alpha+\cos \beta=2 \cos \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)

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=\tan \alpha+2 \tan 2\alpha +4 \tan 4 \alpha+8 \tan 8 \alpha+ 16 \cot 16 \alpha \\=\tan \alpha+2 \tan 2\alpha +4 \tan 4 \alpha+8 \tan 8 \alpha+16 (\frac{1- \tan^2 8 \alpha}{ 2\tan 8 \alpha})\\=\tan \alpha+2 \tan 2\alpha +4 \tan 4 \alpha+\frac{8 \tan^2 8 \alpha+8- 8\tan^2 8 \alpha}{\tan 8\alpha}\\ =\tan \alpha+2 \tan 2\alpha +4 \tan 4 \alpha+8 \cot8 \alpha\\ =\tan \alpha+2 \tan 2\alpha +4 \tan 4 \alpha+8 (\frac{1- \tan^2 4 \alpha}{ 2\tan 4 \alpha}) \\=\tan \alpha+2 \tan 2\alpha +\frac{4 \tan^2 4 \alpha+4- 4\tan^2 4 \alpha}{\tan 4\alpha}\\ =\tan \alpha+2 \tan 2\alpha +4 \cot 4 \alpha\\ =\tan \alpha+2 \tan 2\alpha +4 (\frac{1- \tan^2 2 \alpha}{ 2\tan 2 \alpha})\\ =\tan \alpha+\frac{2 \tan^2 2 \alpha+2- 2\tan^2 4 \alpha}{\tan 2\alpha}\\ =\tan \alpha+2 \cot 2\alpha \\=\tan \alpha+2 (\frac{1- \tan^2 \alpha}{ 2\tan \alpha})\\ =cot \alpha

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Ritika Kankaria

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