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The numerical value of tan ( {2{{ }^{ - 1}}{1}{5} - { }{4}} )

Question

The numerical value of tan $(2 tan^{- 1} \frac{1}{5} - \frac{π}{4})$ is

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The numerical value of tan

(2 tan^{- 1} \frac{1}{5} - \frac{π}{4})

is

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Text solutionVerified

Solution :-

Let

A = t an [2 t a n^{- 1} (\frac{1}{5}) - \frac{π}{4}]

Let

2 t a n^{- 1} (\frac{1}{5}) = θ

∴ A = t an (θ - π /4) = \frac{t an θ - t anπ /4}{1 + t an θ t anπ /4} = \frac{t an θ - 1}{t an θ + 1}

\frac{θ}{2} = \frac{2 t a n ^{- 1}}{2} (\frac{1}{5}) = t a n^{- 1} (\frac{1}{5}) = \frac{θ}{2} \Rightarrow t an \frac{θ}{2} = \frac{1}{5}

But

t an θ = \frac{2 t an θ /2}{1 - t a n ^{2} θ /2} = \frac{2 \cdot 1/5}{1 - 1/25} = \frac{2/5}{24/25} = \frac{5}{12}

∴ A = \frac{t an θ - 1}{t an θ + 1} = \frac{5/12 - 1}{5/12 + 1} = \frac{- 7}{17}

∴

Numerical Value =

∣ ∣ \frac{- 7}{17} ∣ ∣ = 7/17

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Question Text	The numerical value of tan $(2 tan^{- 1} \frac{1}{5} - \frac{π}{4})$ is
Answer Type	Text solution:1
Upvotes	150