Question

log(tan5°)+log(tan25°)+log(tan65°)+log(85°)=

give me full explanation.
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Answer 1

Before solving the question,

tan (θ) = cot ( 90 - θ)

So,

⇒tan 85 = cot ( 90 - 85) = cot5

⇒tan65 = cot (90-65)= cot 25

Also,

tanθ. cotθ = 1

Now let's get into the question,

⇒log(tan5°)+log(tan25°)+log(tan65°)+log(tan85°)

Let's use laws of logarithms, which says log a + log b = log ab

⇒log ( tan5°. tan25°. tan65°. tan85°)

⇒ log ( tan5°. tan25°. cot(90-65°). cot(90-85°))

⇒log ( tan5°. tan25°. cot5°. cot25°)

⇒log (( tan5°. cot5°) . (tan25°. cot25°))

⇒log ( 1 × 1)

⇒log 1

⇒ 0 ( Final answer)

We know that, Anything powered to 0 gives 1. Hence, Log 1 = 0.

Therefore, log(tan5°)+log(tan25°)+log(tan65°)+log(tan85°) = 0