Let's evaluate the given trigonometric expression step-by-step. The expression is: \[ \rm \frac{tan^230^\circ}{sec^230^\circ}+\frac{cosec^245^\circ}{cot^245^\circ}-\frac{sec^260^\circ}{cosec^260^\circ} \] To evaluate this expression, we need the standard trigonometric values for the angles 30°, 45°, and 60°.

Standard Trigonometric Values for 30°, 45°, and 60°

Here are the required standard trigonometric values:

Now, let's find the squares of the required values:

• \(\tan^2 30^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3}\)
• \(\sec^2 30^\circ = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3}\)
• \(\operatorname{cosec}^2 45^\circ = (\sqrt{2})^2 = 2\)
• \(\cot^2 45^\circ = (1)^2 = 1\)
• \(\sec^2 60^\circ = (2)^2 = 4\)
• \(\operatorname{cosec}^2 60^\circ = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3}\)

Method 1: Direct Substitution of Values

Substitute these squared values back into the original expression:

Now, simplify each term:

• First term: \( \frac{\frac{1}{3}}{\frac{4}{3}} = \frac{1}{3} \times \frac{3}{4} = \frac{1}{4} \)
• Second term: \( \frac{2}{1} = 2 \)
• Third term: \( \frac{4}{\frac{4}{3}} = 4 \times \frac{3}{4} = 3 \)

Substitute the simplified terms back into the expression:

Method 2: Using Trigonometric Identities

We can also simplify the expression using trigonometric identities first. Let's look at the terms:

Term 1: \( \frac{\tan^2\theta}{\sec^2\theta} \) Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \). So, \( \frac{\tan^2\theta}{\sec^2\theta} = \frac{(\sin \theta / \cos \theta)^2}{(1 / \cos \theta)^2} = \frac{\sin^2 \theta / \cos^2 \theta}{1 / \cos^2 \theta} = \frac{\sin^2 \theta}{\cos^2 \theta} \times \cos^2 \theta = \sin^2 \theta \). Thus, \( \frac{\tan^230^\circ}{\sec^230^\circ} = \sin^2 30^\circ \).

Term 2: \( \frac{\operatorname{cosec}^2\theta}{\cot^2\theta} \) Recall that \( \operatorname{cosec} \theta = \frac{1}{\sin \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). So, \( \frac{\operatorname{cosec}^2\theta}{\cot^2\theta} = \frac{(1 / \sin \theta)^2}{(\cos \theta / \sin \theta)^2} = \frac{1 / \sin^2 \theta}{\cos^2 \theta / \sin^2 \theta} = \frac{1}{\sin^2 \theta} \times \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta} = \sec^2 \theta \). Alternatively, using \( \operatorname{cosec}^2 \theta = 1 + \cot^2 \theta \), \( \frac{\operatorname{cosec}^2\theta}{\cot^2\theta} = \frac{1 + \cot^2 \theta}{\cot^2 \theta} = \frac{1}{\cot^2 \theta} + 1 = \tan^2 \theta + 1 = \sec^2 \theta \). Thus, \( \frac{\operatorname{cosec}^245^\circ}{\cot^245^\circ} = \sec^2 45^\circ \).

Term 3: \( \frac{\sec^2\theta}{\operatorname{cosec}^2\theta} \) Recall that \( \sec \theta = \frac{1}{\cos \theta} \) and \( \operatorname{cosec} \theta = \frac{1}{\sin \theta} \). So, \( \frac{\sec^2\theta}{\operatorname{cosec}^2\theta} = \frac{(1 / \cos \theta)^2}{(1 / \sin \theta)^2} = \frac{1 / \cos^2 \theta}{1 / \sin^2 \theta} = \frac{1}{\cos^2 \theta} \times \sin^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} = \tan^2 \theta \). Thus, \( \frac{\sec^260^\circ}{\operatorname{cosec}^260^\circ} = \tan^2 60^\circ \).

Using these identities, the expression simplifies to:

Now substitute the standard values:

• \(\sin 30^\circ = \frac{1}{2} \implies \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\)
• \(\sec 45^\circ = \sqrt{2} \implies \sec^2 45^\circ = (\sqrt{2})^2 = 2\)
• \(\tan 60^\circ = \sqrt{3} \implies \tan^2 60^\circ = (\sqrt{3})^2 = 3\)

Substitute these squared values:

Both methods yield the same result. The value of the given trigonometric expression is \(-\frac{3}{4}\).

Revision Table: Key Trigonometric Concepts

Additional Information on Evaluating Trigonometric Expressions

Evaluating trigonometric expressions often requires a strong understanding of the standard trigonometric values for common angles (0°, 30°, 45°, 60°, 90°) and the fundamental trigonometric identities. When faced with such a problem, you can either substitute the numerical values directly at the beginning or use identities to simplify the expression before substituting values. Choosing the right approach depends on the complexity of the expression. For expressions involving squares of ratios, using identities can sometimes simplify the calculation process significantly.

Remember to square the trigonometric values correctly when the function itself is squared (e.g., \(\sin^2 \theta = (\sin \theta)^2\)). Pay attention to the signs when combining terms, especially with subtraction involved.

The ability to recall or quickly derive the standard angle values and apply trigonometric identities is crucial for solving such problems efficiently in exams.