If sin = 1 and sin= 1 2 , then tan (2) tan (2) is equal to

Question

If $\sin (α + β) = 1$ and $\sin (α - β) = \frac{1}{2},$ then $\tan (α + 2 β) \tan (2 α + β)$ is equal to

Smart Achievers · Accepted Answer

(1)

Since, $\sin (α + β) = 1$

$∴$ $α + β = \frac{π}{2}$ $\dots (i)$

and $\sin (α - β) = \frac{1}{2}$

$\Rightarrow$ $α - β = \frac{π}{6}$ $\dots (i i)$

On solving Eqs. (i) and (ii), we get

$α = \frac{π}{3}$ and $β = \frac{π}{6}$

Now, $\tan (α + 2 β) \tan (2 α + β)$

$= \tan (\frac{2 π}{3})$ $\tan (\frac{5 π}{6})$

$= \tan (π - \frac{π}{3})$ $\tan (π - \frac{π}{6})$

$= (- \cot \frac{π}{3})$ $(- \cot \frac{π}{6})$

$= \frac{1}{\sqrt{3}} \times \sqrt{3} = 1$

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