Solving the Trigonometry Product: tan 31° to tan 59°

The problem asks for the value of the product of tangent functions for angles in an arithmetic progression: $\tan 31^\circ \tan 33^\circ \tan 35^\circ \dots \tan 57^\circ \tan 59^\circ$.

The angles are $31^\circ, 33^\circ, 35^\circ, \dots, 57^\circ, 59^\circ$. This is an arithmetic progression with a common difference of $2^\circ$.

Let's identify the angles in the sequence. Starting from $31^\circ$ and adding $2^\circ$ repeatedly until $59^\circ$. The angles are:

• $31^\circ$
• $33^\circ$
• $35^\circ$
• $37^\circ$
• $39^\circ$
• $41^\circ$
• $43^\circ$
• $45^\circ$
• $47^\circ$
• $49^\circ$
• $51^\circ$
• $53^\circ$
• $55^\circ$
• $57^\circ$
• $59^\circ$

There are a total of 15 terms in this product.

Using Complementary Angle Identities for Tangent Product

We can use the complementary angle identity for tangent: $\tan (90^\circ - \theta) = \cot \theta$. Also, recall that $\cot \theta = \frac{1}{\tan \theta}$. Therefore, $\tan \theta \times \tan (90^\circ - \theta) = \tan \theta \times \cot \theta = 1$.

Let's look at the angles in the product. Notice that pairs of angles add up to $90^\circ$:

• $31^\circ + 59^\circ = 90^\circ$
• $33^\circ + 57^\circ = 90^\circ$
• $35^\circ + 55^\circ = 90^\circ$
• ... and so on.

The product is: $\tan 31^\circ \times \tan 33^\circ \times \dots \times \tan 57^\circ \times \tan 59^\circ$.

We can rewrite the terms using the complementary angle identity:

• $\tan 59^\circ = \tan (90^\circ - 31^\circ) = \cot 31^\circ = \frac{1}{\tan 31^\circ}$
• $\tan 57^\circ = \tan (90^\circ - 33^\circ) = \cot 33^\circ = \frac{1}{\tan 33^\circ}$
• $\tan 55^\circ = \tan (90^\circ - 35^\circ) = \cot 35^\circ = \frac{1}{\tan 35^\circ}$
• ...

Pairing Terms and Simplifying the Product

Let the product be $P$. We can group the terms into pairs where the angles sum to $90^\circ$. Since there are 15 terms, the middle term is the $(15+1)/2 = 8$th term, which is $\tan 45^\circ$. The other $15-1 = 14$ terms can be grouped into $14/2 = 7$ pairs.

The product can be written as:

Using the identity $\tan \theta \times \tan (90^\circ - \theta) = 1$ for each pair:

• $\tan 31^\circ \tan 59^\circ = \tan 31^\circ \cot 31^\circ = 1$
• $\tan 33^\circ \tan 57^\circ = \tan 33^\circ \cot 33^\circ = 1$
• $\tan 35^\circ \tan 55^\circ = \tan 35^\circ \cot 35^\circ = 1$
• $\tan 37^\circ \tan 53^\circ = \tan 37^\circ \cot 37^\circ = 1$
• $\tan 39^\circ \tan 51^\circ = \tan 39^\circ \cot 39^\circ = 1$
• $\tan 41^\circ \tan 49^\circ = \tan 41^\circ \cot 41^\circ = 1$
• $\tan 43^\circ \tan 47^\circ = \tan 43^\circ \cot 43^\circ = 1$

The product $P$ becomes:

We know that the value of $\tan 45^\circ$ is 1.

Therefore, the value of the given product $\tan 31^\circ \tan 33^\circ \tan 35^\circ \dots \tan 57^\circ \tan 59^\circ$ is 1.

Revision Table: Key Trigonometry Concepts

Reviewing the key concepts used in this problem:

Additional Information: Trigonometric Products and Series

Problems involving products of trigonometric functions often utilize complementary or supplementary angle identities. When angles form an arithmetic progression, look for patterns where terms can be paired up.

• Complementary Angles ($90^\circ$): Identities like $\sin(90^\circ - \theta) = \cos \theta$, $\cos(90^\circ - \theta) = \sin \theta$, $\tan(90^\circ - \theta) = \cot \theta$.
• Supplementary Angles ($180^\circ$): Identities like $\sin(180^\circ - \theta) = \sin \theta$, $\cos(180^\circ - \theta) = -\cos \theta$, $\tan(180^\circ - \theta) = -\tan \theta$.
• Angles summing to $180^\circ$: For a product of tangents like $\tan A \tan B \tan C \dots$, if you find pairs where $A+B = 180^\circ$, $\tan A + \tan B = \tan A + \tan(180^\circ - A) = \tan A - \tan A = 0$ (for sum) or $\tan A \tan B = \tan A \tan(180^\circ - A) = \tan A (-\tan A) = -\tan^2 A$ (for product, less useful for product simplification). The $90^\circ$ sum is more powerful for simplifying products to 1.
• General Approach: For a product of tangents with angles in AP, check if pairs of angles sum to $90^\circ$. If they do, most terms will cancel out to 1, leaving potentially a middle term like $\tan 45^\circ$ or $\tan 90^\circ$ (which is undefined) or $\tan \theta$ where $\theta$ is half of the sum.