Finding the Value of a Trigonometric Product

The problem asks us to find the value of the expression cot 25° cot 35° cot 45° cot 55° cot 65°. This involves evaluating a product of cotangent values at specific angles.

Understanding the Cotangent Function

The cotangent function, denoted as cot θ, is the reciprocal of the tangent function, i.e., cot θ = \(\frac{1}{\tan \theta}\) or cot θ = \(\frac{\cos \theta}{\sin \theta}\).

Using Trigonometric Identities to Simplify

We can simplify this expression by using the relationship between cotangent and tangent for complementary angles. The identity is: cot θ = tan (90° - θ).

Let's look at the angles in the given product:

• 25°
• 35°
• 45°
• 55°
• 65°

Notice that some pairs of angles add up to 90° (complementary angles):

• 25° + 65° = 90°
• 35° + 55° = 90°

Using the identity cot θ = tan (90° - θ), we can rewrite some terms:

• cot 65° = cot (90° - 25°) = tan 25°
• cot 55° = cot (90° - 35°) = tan 35°

Also, we know the exact value of cot 45°.

Evaluating Cotangent at 45 Degrees

The value of cot 45° is well-known. Since cot 45° = \(\frac{1}{\tan 45^\circ}\) and tan 45° = 1, we have cot 45° = \(\frac{1}{1}\) = 1.

Substituting and Simplifying the Expression

Now, let's substitute the rewritten terms and the value of cot 45° back into the original expression:

Original expression: cot 25° cot 35° cot 45° cot 55° cot 65°

Substitute cot 55° = tan 35° and cot 65° = tan 25°:

cot 25° cot 35° cot 45° (tan 35°) (tan 25°)

Rearrange the terms to group complementary angles:

(cot 25° tan 25°) (cot 35° tan 35°) cot 45°

Now, we use another important trigonometric identity: cot θ ⋅ tan θ = 1. This is because tan θ = \(\frac{1}{\cot \theta}\), so cot θ ⋅ \(\frac{1}{\cot \theta}\) = 1 (for \(\theta \neq n\pi/2\)).

Apply this identity to the grouped terms:

• cot 25° tan 25° = 1
• cot 35° tan 35° = 1

Substitute these values back into the expression:

(1) ⋅ (1) ⋅ cot 45°

Finally, substitute the value of cot 45° = 1:

1 ⋅ 1 ⋅ 1 = 1

Thus, the value of cot 25° cot 35° cot 45° cot 55° cot 65° is 1.

Step-by-Step Calculation Summary

1. Identify complementary angle pairs: (25°, 65°) and (35°, 55°).
2. Use cot θ = tan (90° - θ) to rewrite cot 65° as tan 25° and cot 55° as tan 35°.
3. Rewrite the expression: cot 25° cot 35° cot 45° (tan 35°) (tan 25°).
4. Rearrange terms: (cot 25° tan 25°) (cot 35° tan 35°) cot 45°.
5. Use cot θ tan θ = 1: (1) ⋅ (1) ⋅ cot 45°.
6. Use cot 45° = 1: 1 ⋅ 1 ⋅ 1 = 1.

The final value is 1.

Revision Table: Trigonometric Identities and Values

Additional Information: Complementary Angles in Trigonometry

Complementary angles are two angles that add up to 90 degrees. The relationships between trigonometric functions of complementary angles are very useful for simplifying expressions and solving problems. For example:

• sin (90° - θ) = cos θ
• cos (90° - θ) = sin θ
• tan (90° - θ) = cot θ
• cot (90° - θ) = tan θ
• sec (90° - θ) = cosec θ
• cosec (90° - θ) = sec θ

These identities show that the sine of an angle is equal to the cosine of its complement, the tangent of an angle is equal to the cotangent of its complement, and so on. This property was key to simplifying the given product involving cot 25°, cot 35°, cot 45°, cot 55°, and cot 65°.