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What is the value of $tan 15 ∘$ ?

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2 + 3

2 − 3

3 − 2

1 2 − 5

\tan 15 ⁰ = \tan (45 ⁰ - 30 ⁰)

answer is B.

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Detailed Solution

We know that,

\tan 15 ⁰ = \tan (45 ⁰ - 30 ⁰)

… (1)
By using

tan (A − B) = tan A − tan B 1 + (tan A × tan B)

, we get

tan (45 ∘ − 30 ∘) = tan 45 ∘ − tan 30 ∘ 1 + (tan 45 ∘ × tan 30 ∘)

Substitute the values

tan 45 ∘ = 1, tan 30 ∘ = 13

in the above expression, we get

tan 15 ∘ = 1 − 13 1 + 1 × 13 = 3 − 1 3 1 + 13 = 3 − 1 3 3 + 1 3

tan 15 ∘ = 3 − 1 3 + 1

tan 15 ∘ = 3 − 1 3 + 1 × 3 − 1 3 − 1 = 3 − 1 2 3 + 1 × 3 − 1

By using the formulae

(a − b) 2 = a 2 + b 2 − 2 a b, (a − b) (a + b) = a 2 − b 2

, we get

tan 15 ∘ = 32 + 12 − 2 (3) (1) 32 − 12 = 3 + 1 − 23 3 − 1 = 4 − 23 2 = 2 − 3

Hence, the correct option is 2.

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