Understanding the Trigonometry Problem

The question asks us to find the numerical value of a given trigonometric expression. The expression involves various trigonometric functions like tangent, cosecant, and cosine, with different angles. To solve this, we need to simplify both the numerator and the denominator using trigonometric identities, especially complementary angle identities and Pythagorean identities.

Step-by-Step Simplification of the Expression

Let's break down the expression into the numerator and the denominator and simplify each part separately.

Simplifying the Numerator

The numerator is \( \tan9^\circ \tan23^\circ \tan60^\circ \tan67^\circ \tan81^\circ \). We can use the complementary angle identity: \( \tan(90^\circ - \theta) = \cot\theta = \frac{1}{\tan\theta} \).

• \( \tan81^\circ = \tan(90^\circ - 9^\circ) = \cot9^\circ = \frac{1}{\tan9^\circ} \)
• \( \tan67^\circ = \tan(90^\circ - 23^\circ) = \cot23^\circ = \frac{1}{\tan23^\circ} \)

Substitute these back into the numerator:

\( \text{Numerator} = \tan9^\circ \tan23^\circ \tan60^\circ \left(\frac{1}{\tan23^\circ}\right) \left(\frac{1}{\tan9^\circ}\right) \)

The \( \tan9^\circ \) terms cancel out, and the \( \tan23^\circ \) terms cancel out.

\( \text{Numerator} = \tan60^\circ \)

We know the standard value of \( \tan60^\circ \).

\( \tan60^\circ = \sqrt{3} \)

So, the numerator simplifies to \( \sqrt{3} \).

Simplifying the Denominator

The denominator is \( \operatorname{cosec}^2{72}^\circ + {{\cos }^2}15^\circ - {{\tan }^2}18^\circ + {{\cos }^2}75^\circ \). We can use complementary angle identities and Pythagorean identities.

• Complementary angle identity: \( \operatorname{cosec}(90^\circ - \theta) = \sec\theta \)
• \( \operatorname{cosec}^2 72^\circ = \operatorname{cosec}^2 (90^\circ - 18^\circ) = \sec^2 18^\circ \)
• Complementary angle identity: \( \cos(90^\circ - \theta) = \sin\theta \)
• \( \cos^2 75^\circ = \cos^2 (90^\circ - 15^\circ) = \sin^2 15^\circ \)

Substitute these back into the denominator:

\( \text{Denominator} = \sec^2 18^\circ + \cos^2 15^\circ - \tan^2 18^\circ + \sin^2 15^\circ \)

Rearrange the terms to group related functions and angles:

\( \text{Denominator} = (\sec^2 18^\circ - \tan^2 18^\circ) + (\cos^2 15^\circ + \sin^2 15^\circ) \)

Now, we use the Pythagorean identities:

• \( \sec^2 \theta - \tan^2 \theta = 1 \)
• \( \cos^2 \theta + \sin^2 \theta = 1 \)

Applying these identities to the grouped terms:

• \( \sec^2 18^\circ - \tan^2 18^\circ = 1 \) (with \( \theta = 18^\circ \))
• \( \cos^2 15^\circ + \sin^2 15^\circ = 1 \) (with \( \theta = 15^\circ \))

So, the denominator simplifies to:

\( \text{Denominator} = 1 + 1 = 2 \)

Calculating the Final Value

The original expression is the numerator divided by the denominator:

\( \frac{\text{Numerator}}{\text{Denominator}} = \frac{\sqrt{3}}{2} \)

Thus, the value of the given trigonometric expression is \( \frac{\sqrt{3}}{2} \).

Summary of Simplification Steps

• Simplified the numerator \( \tan9^\circ \tan23^\circ \tan60^\circ \tan67^\circ \tan81^\circ \) using complementary angle identities and the value of \( \tan60^\circ \) to get \( \sqrt{3} \).
• Simplified the denominator \( \operatorname{cosec}^2{72}^\circ + {{\cos }^2}15^\circ - {{\tan }^2}18^\circ + {{\cos }^2}75^\circ \) using complementary angle identities and Pythagorean identities (\( \sec^2 \theta - \tan^2 \theta = 1 \) and \( \cos^2 \theta + \sin^2 \theta = 1 \)) to get \( 2 \).
• Divided the simplified numerator by the simplified denominator to find the final value \( \frac{\sqrt{3}}{2} \).

Revision Table: Trigonometric Expression Evaluation

Additional Information: Mastering Trigonometric Identities

Trigonometric identities are fundamental equations that are true for all values of the variables involved. Mastering these identities is crucial for simplifying trigonometric expressions and solving trigonometric equations. Here are some key points:

• Reciprocal Identities: Relate pairs of functions (e.g., \( \sin\theta = 1/\operatorname{cosec}\theta \)).
• Quotient Identities: Express tangent and cotangent in terms of sine and cosine (e.g., \( \tan\theta = \sin\theta/\cos\theta \)).
• Pythagorean Identities: Derived from the Pythagorean theorem, relating squares of functions (e.g., \( \sin^2\theta + \cos^2\theta = 1 \)). The identities \( \sec^2\theta - \tan^2\theta = 1 \) and \( \operatorname{cosec}^2\theta - \cot^2\theta = 1 \) are also derived from this.
• Complementary Angle Identities: Relate function values of an angle to those of its complement (\( 90^\circ - \theta \)). These are particularly useful when you see pairs of angles that add up to \( 90^\circ \), like \( 9^\circ \) and \( 81^\circ \), or \( 23^\circ \) and \( 67^\circ \), or \( 18^\circ \) and \( 72^\circ \), or \( 15^\circ \) and \( 75^\circ \) in this problem.
• Practice: Regularly solving problems that require identity manipulation is the best way to become proficient. Look for patterns in the angles and functions within the expression.