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Trigonometric Ratios of Multiple and Submultiple Angles — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Ratios of Multiple and Submultiple Angles2 Marks
Question
Prove that: $\tan\frac{\pi}{12}+\tan\frac{\pi}{6}+\tan\frac{\pi}{12}\tan\frac{\pi}{6}=1$
✓

Answer

We have, $45^\circ=30^\circ+15^\circ$ $\Rightarrow\tan45^\circ=\tan(30^\circ+15^\circ)$ $\Rightarrow1=\frac{\tan30^\circ+\tan15^\circ}{1-\tan30^\circ\tan15^\circ}$ $\Big[\because\tan\text{(A}+\text{B)}=\frac{\tan\text{A}+\tan\text{B}}{1-\tan\text{A}\tan\text{B}}\Big]$ $\Rightarrow 1-\tan30^\circ\tan30^\circ=\tan15^\circ+\tan30^\circ$ $\Rightarrow 1=\tan15^\circ+\tan30^\circ+\tan30^\circ\tan15^\circ$ $\Rightarrow \tan15^\circ+\tan30^\circ+\tan15^\circ\tan30^\circ=1$ Hence proved.

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