Evaluating Trigonometric Expressions

We need to find the value of the given trigonometric expression:

\(\rm \frac{(cos\,9^\circ+sin \,81^\circ)(sec\,9^\circ+cosec\,81^\circ)}{cosec^2\,71^\circ+cos^2\,15^\circ-tan^2\,19^\circ+cos^2\,75^\circ} \)

Let's simplify the numerator and the denominator separately using trigonometric identities.

Simplifying the Numerator

The numerator is \((\cos\,9^\circ+\sin\,81^\circ)(\sec\,9^\circ+\cosec\,81^\circ)\).

We can use the complementary angle identities:

• \(\sin(90^\circ - x) = \cos x\)
• \(\cosec(90^\circ - x) = \sec x\)

Applying these identities:

• \(\sin\,81^\circ = \sin(90^\circ - 9^\circ) = \cos\,9^\circ\)
• \(\cosec\,81^\circ = \cosec(90^\circ - 9^\circ) = \sec\,9^\circ\)

Substitute these values back into the numerator expression:

\((\cos\,9^\circ + \cos\,9^\circ)(\sec\,9^\circ + \sec\,9^\circ)\)

This simplifies to:

\((2\,\cos\,9^\circ)(2\,\sec\,9^\circ)\)

\(= 4\,\cos\,9^\circ\,\sec\,9^\circ\)

Using the reciprocal identity \(\sec x = \frac{1}{\cos x}\):

\(= 4\,\cos\,9^\circ\,\left(\frac{1}{\cos\,9^\circ}\right)\)

\(= 4 \)

So, the value of the numerator is 4.

Simplifying the Denominator

The denominator is \(\cosec^2\,71^\circ+\cos^2\,15^\circ-\tan^2\,19^\circ+\cos^2\,75^\circ\).

Again, we use complementary angle identities:

• \(\cosec(90^\circ - x) = \sec x\)
• \(\cos(90^\circ - x) = \sin x\)

Applying these identities:

• \(\cosec\,71^\circ = \cosec(90^\circ - 19^\circ) = \sec\,19^\circ\)
• \(\cos\,15^\circ = \cos(90^\circ - 75^\circ) = \sin\,75^\circ\)

Substitute these into the denominator expression:

\((\sec\,19^\circ)^2 + (\sin\,75^\circ)^2 - \tan^2\,19^\circ + \cos^2\,75^\circ\)

\(= \sec^2\,19^\circ + \sin^2\,75^\circ - \tan^2\,19^\circ + \cos^2\,75^\circ\)

Rearrange the terms to group identities:

\(= (\sec^2\,19^\circ - \tan^2\,19^\circ) + (\sin^2\,75^\circ + \cos^2\,75^\circ)\)

We use the following Pythagorean identities:

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

Applying these identities:

• \(\sec^2\,19^\circ - \tan^2\,19^\circ = 1\)
• \(\sin^2\,75^\circ + \cos^2\,75^\circ = 1\)

Substitute these values back into the denominator expression:

\(= 1 + 1\)

\(= 2\)

So, the value of the denominator is 2.

Calculating the Final Value

The original expression is \(\rm \frac{Numerator}{Denominator}\).

Substituting the simplified values:

\(\rm \frac{4}{2} = 2\)

Thus, the value of the given expression is 2.

Revision Table: Key Trigonometric Identities

Additional Information on Trigonometric Concepts

Complementary Angles: Two angles are called complementary if their sum is \(90^\circ\). For example, \(9^\circ\) and \(81^\circ\) are complementary because \(9^\circ + 81^\circ = 90^\circ\). Similarly, \(19^\circ\) and \(71^\circ\) are complementary, and \(15^\circ\) and \(75^\circ\) are complementary.

The relationships between trigonometric functions of complementary angles are very useful in simplifying expressions. As used in the solution:

• \(\sin(90^\circ - x) = \cos x\)
• \(\cos(90^\circ - x) = \sin x\)
• \(\tan(90^\circ - x) = \cot x\)
• \(\cot(90^\circ - x) = \tan x\)
• \(\sec(90^\circ - x) = \cosec x\)
• \(\cosec(90^\circ - x) = \sec x\)

Pythagorean Identities: These are fundamental identities derived from the Pythagorean theorem applied to a right-angled triangle. The three main Pythagorean identities are:

• \(\sin^2 x + \cos^2 x = 1\)
• \(1 + \tan^2 x = \sec^2 x\) (or \(\sec^2 x - \tan^2 x = 1\))
• \(1 + \cot^2 x = \cosec^2 x\) (or \(\cosec^2 x - \cot^2 x = 1\))

These identities are crucial for simplifying expressions and proving other trigonometric relationships.

By combining the use of complementary angle identities and Pythagorean identities, complex trigonometric expressions can often be simplified to a numerical value or a much simpler expression.