What is cot 1° cot 23° cot 45° cot 67° cot 89° equal to?
1
Understanding the Trigonometric Product Question
The question asks us to find the value of the product of several cotangent terms: \(\cot 1^\circ \cot 23^\circ \cot 45^\circ \cot 67^\circ \cot 89^\circ\). To solve this, we need to use some fundamental trigonometric identities and properties related to complementary angles.
Key Trigonometric Identities and Concepts
We will use the following trigonometric concepts:
- The value of \(\cot 45^\circ\).
- The relationship between cotangent and tangent: \(\cot \theta = \frac{1}{\tan \theta}\).
- The complementary angle identity: \(\cot (90^\circ - \theta) = \tan \theta\).
Step-by-Step Evaluation of the Product
Let's look at the angles given in the product: \(1^\circ, 23^\circ, 45^\circ, 67^\circ, 89^\circ\). We can observe pairs of angles that add up to \(90^\circ\) (complementary angles):
- \(1^\circ + 89^\circ = 90^\circ\)
- \(23^\circ + 67^\circ = 90^\circ\)
- \(45^\circ\) is a special angle on its own.
Now, let's apply the complementary angle identity \(\cot (90^\circ - \theta) = \tan \theta\) to the angles greater than \(45^\circ\):
- \(\cot 89^\circ = \cot (90^\circ - 1^\circ) = \tan 1^\circ\)
- \(\cot 67^\circ = \cot (90^\circ - 23^\circ) = \tan 23^\circ\)
We also know the exact value of \(\cot 45^\circ\):
- \(\cot 45^\circ = 1\)
Now, substitute these values back into the original product expression:
Original product = \(\cot 1^\circ \cot 23^\circ \cot 45^\circ \cot 67^\circ \cot 89^\circ\)
Substitute the transformed terms:
Product = \(\cot 1^\circ \cot 23^\circ (1) (\tan 23^\circ) (\tan 1^\circ)\)
Rearrange the terms to group the complementary pairs:
Product = \((\cot 1^\circ \tan 1^\circ) (\cot 23^\circ \tan 23^\circ) (\cot 45^\circ)\)
Using the identity \(\cot \theta = \frac{1}{\tan \theta}\), we know that \(\cot \theta \tan \theta = \left(\frac{1}{\tan \theta}\right) \tan \theta = 1\) (as long as \(\tan \theta \neq 0\)).
Applying this identity to the pairs:
- \(\cot 1^\circ \tan 1^\circ = 1\)
- \(\cot 23^\circ \tan 23^\circ = 1\)
Now, substitute these values back into the expression:
Product = \((1) \times (1) \times (\cot 45^\circ)\)
Since \(\cot 45^\circ = 1\):
Product = \(1 \times 1 \times 1 = 1\)
So, the value of \(\cot 1^\circ \cot 23^\circ \cot 45^\circ \cot 67^\circ \cot 89^\circ\) is \(1\).
Summary of the Calculation
We used the property of complementary angles and the reciprocal identity of cotangent and tangent to simplify the product. Pairing \(\cot \theta\) with \(\cot (90^\circ - \theta)\) which equals \(\tan \theta\), allowed terms to cancel out to 1.
| Term | Complementary Angle | Relation | Product Pair | Value |
|---|---|---|---|---|
| \(\cot 1^\circ\) | \(\cot 89^\circ = \tan 1^\circ\) | \(\cot \theta \tan \theta = 1\) | \(\cot 1^\circ \cot 89^\circ\) | \(\cot 1^\circ \tan 1^\circ = 1\) |
| \(\cot 23^\circ\) | \(\cot 67^\circ = \tan 23^\circ\) | \(\cot \theta \tan \theta = 1\) | \(\cot 23^\circ \cot 67^\circ\) | \(\cot 23^\circ \tan 23^\circ = 1\) |
| \(\cot 45^\circ\) | - | Special angle | \(\cot 45^\circ\) | \(1\) |
Total product = \((\cot 1^\circ \cot 89^\circ) \times (\cot 23^\circ \cot 67^\circ) \times \cot 45^\circ = 1 \times 1 \times 1 = 1\).
Revision Table: Key Trigonometric Identities
| Identity | Description |
|---|---|
| \(\cot 45^\circ = 1\) | Value of cotangent at 45 degrees. |
| \(\cot \theta = \frac{\cos \theta}{\sin \theta}\) | Ratio definition of cotangent. |
| \(\cot \theta = \frac{1}{\tan \theta}\) | Reciprocal identity with tangent. |
| \(\cot (90^\circ - \theta) = \tan \theta\) | Complementary angle identity for cotangent. |
| \(\tan (90^\circ - \theta) = \cot \theta\) | Complementary angle identity for tangent. |
| \(\cot \theta \tan \theta = 1\) | Product of cotangent and tangent for the same angle. |
Additional Information: Products of Trigonometric Ratios
Problems involving products of trigonometric ratios of angles often simplify nicely when the angles are related through complementary or supplementary angle identities. Recognising pairs of angles that add up to \(90^\circ\) or \(180^\circ\) is a common strategy.
For a product of \(\tan\) or \(\cot\) terms with angles in an arithmetic progression from \(1^\circ\) to \(89^\circ\) (like \(\tan 1^\circ \tan 2^\circ \dots \tan 89^\circ\)), the product will often simplify to 1. This is because \(\tan \theta \tan (90^\circ - \theta) = \tan \theta \cot \theta = 1\). Most terms pair up, and the middle term (\(\tan 45^\circ\) or \(\cot 45^\circ\)) is 1.
In this specific problem, the angles were not consecutive but were carefully chosen to form complementary pairs (\(1^\circ\) with \(89^\circ\), \(23^\circ\) with \(67^\circ\)) plus the special angle \(45^\circ\).
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