log

Algebra Level 2

log ⁡ ( tan ⁡ 1 ∘ ) + log ⁡ ( tan ⁡ 2 ∘ ) + log ⁡ ( tan ⁡ 3 ∘ ) + . . . . . . + log ⁡ ( tan ⁡ 8 9 ∘ ) \log\left( \tan 1^{\circ }\right) + \log\left( \tan 2^{\circ }\right) +\log\left( \tan 3^{\circ }\right) + ......+ \log\left( \tan 89^{\circ }\right)

If the above expression is equal to x x , then find the value of x + 5 x+5 .


The answer is 5.

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Mahmoud Gadelrab by Mahmoud Gadelrab , 23, Egypt

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1 solution

Most of us know that...

log ⁡ a + log ⁡ b = log ⁡ a b \log a + \log b = \log ab

and that...

tan ⁡ θ × tan ⁡ ( θ − 90 ) = 1 \tan \theta \times \tan (\theta - 90) = 1

and tan ⁡ 45 = 1 \tan 45 = 1

Simplifying the equation, we get...

log ⁡ tan ⁡ ( 1 × 2 × 3 × . . . × 44 × 45 × 46 × . . . × 87 × 88 × 89 ) \log \tan (1 \times 2 \times 3 \times ... \times 44 \times 45 \times 46 \times ... \times 87 \times 88 \times 89) log ⁡ tan ⁡ ( 1 × 2 × 3 × . . . × 44 × 45 × ( 90 − 44 ) × . . . × ( 90 − 3 ) × ( 90 − 2 ) × ( 90 − 1 ) \log \tan (1 \times 2 \times 3 \times ... \times 44 \times 45 \times (90 - 44) \times ... \times (90 - 3) \times (90 - 2) \times (90 - 1) log ⁡ 1 \log 1 0 \boxed{0}

So, x + 5 => 5.

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