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Question Number 63363 by rajesh4661kumar@gamil.com last updated on 03/Jul/19

Commented by Prithwish sen last updated on 03/Jul/19

$$\tan 70=\frac{\tan 40+\tan 20+\tan 10-\tan 40\tan 20\tan 10}{1-\tan 40\tan 20-\tan 40\tan 10-\tan 20\tan 10}$$$$\tan 70-\tan 70\tan 40\tan 20-\tan 70\tan 40\tan 10-\tan 70\tan 20\tan 10$$$$=\tan 40+\tan 20+\tan 10-\tan 40\tan 20\tan 10$$$$\because \tan 70\tan 20=1\therefore \tan 70\tan 40\tan 20=\tan 40\mathrm{and}\tan 70\tan 20\tan 10=\tan 10$$$$\Rightarrow \tan 70-2\tan 40-\tan 20=2\tan 10+\tan 40\tan 10(\tan 70-\tan 20)$$$$=2\tan 10+\tan 40\mathrm{tam}10\left(\frac{1-{\tan }^{2}20}{\tan 20}\right)=2\tan 10+\tan 10\left(\frac{2\tan 20}{1-{\tan }^{2}20}\right)\left(\frac{1-{\tan }^{2}20}{\tan 20}\right)=4\tan 10\mathrm{proved}.$$$$$$

Commented by Tony Lin last updated on 03/Jul/19

$$tan70-tan20-2tan40$$$$=\frac{sin70}{cos70}-\frac{sin20}{cos20}-\frac{2sin40}{cos40}$$$$=\frac{sin50}{cos70cos20}-\frac{2sin40}{cos40}$$$$=\frac{sin50cos40-2sin40cos70cos20}{cos70cos20cos40}$$$$=\frac{sin50cos40-sin40(cos90+cos50)}{\frac{1}{2}(cos90+cos50)cos40}$$$$=\frac{sin50cos40-cos50sin40}{\frac{1}{4}(cos90+cos10)}$$$$=\frac{4sin10}{cos10}$$$$=4tan10$$

Commented by Prithwish sen last updated on 03/Jul/19

$$\mathrm{Excellent}\mathrm{sir}$$

Answered by Hope last updated on 03/Jul/19

$$\frac{sin70}{cos70}-\frac{sin20}{cos20}-\frac{2sin40}{cos40}$$$$=\frac{sin(70-20)}{cos70cos20}-\frac{2sin40}{cos40}$$$$=\frac{2sin50}{cos90+cos50}-\frac{2sin40}{cos40}$$$$=\frac{2sin50}{cos50}-\frac{2sin40}{cos40}$$$$=\frac{2(sin50cos40-cos50sin40)}{2cos50cos40}\times 2$$$$=\frac{4sin10}{cos90+cos10}$$$$=4tan10$$$$$$$$$$$$$$

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