Trigo Product

Geometry Level 2

The value of the given expression is : cot ⁡ 10 cot ⁡ 30 cot ⁡ 50 cot ⁡ 70 \cot 10 \cot 30 \cot 50 \cot 70


The answer is 3.

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Danish Ahmed by Danish Ahmed , 24, India

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2 solutions

Danish Ahmed
Jan 4, 2015

cot ⁡ 10 cot ⁡ 30 cot ⁡ 50 cot ⁡ 70 = cot ⁡ ( 90 − 80 ) cot ⁡ 30 cot ⁡ ( 90 − 40 ) cot ⁡ ( 90 − 20 ) = cot ⁡ 30 tan ⁡ 10 tan ⁡ 40 tan ⁡ 80 = cot ⁡ 30 tan ⁡ 20 tan ⁡ ( 60 − 20 ) tan ⁡ ( 60 + 20 ) = cot ⁡ 30 tan ⁡ ( 3 × 20 ) = 3 tan ⁡ 60 = 3 3 = 3 \begin{aligned} & \cot 10 \cot 30 \cot 50 \cot 70 \\ & = \cot (90 - 80) \cot 30 \cot (90 - 40) \cot (90 - 20) \\ & = \cot 30 \tan 10 \tan 40 \tan 80 \\ & = \cot 30 \tan 20 \tan (60 - 20) \tan (60 + 20) \\ & = \cot 30 \tan (3 \times 20) \\ & = \sqrt{3} \tan 60 \\ &= \sqrt{3} \sqrt{3} \\ &= \boxed{3} \end{aligned}

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Samuel Kettlewell
Jan 15, 2015

c o t x = c o s x s i n x cot x = \frac{cos x}{sin x} c o t 10 ⋅ c o t 30 ⋅ c o t 50 ⋅ c o t 70 = c o s 10 ⋅ c o s 30 ⋅ c o s 50 ⋅ c o s 70 s i n 10 ⋅ s i n 30 ⋅ s i n 50 ⋅ s i n 70 = 3 cot 10 \cdot cot 30 \cdot cot 50 \cdot cot 70 = \frac{cos 10 \cdot cos 30 \cdot cos 50 \cdot cos 70}{sin 10 \cdot sin 30 \cdot sin 50 \cdot sin 70}=3

0 Helpful 0 Interesting 0 Brilliant 0 Confused

mention the formula you have applied in the last step please. :)

Rohit Ner - 6 years, 4 months ago

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