Evaluating Trigonometric Expressions: \(\sqrt3\) cosec 20° - sec 20°

We are asked to find the value of the trigonometric expression \(\sqrt3\) cosec 20° - sec 20°.

First, let's rewrite the cosec and sec terms using their definitions in terms of sin and cos:

• cosec \(\theta\) = \(\frac{1}{\sin \theta}\)
• sec \(\theta\) = \(\frac{1}{\cos \theta}\)

Using these definitions, the expression becomes:

Expression = \(\frac{\sqrt3}{\sin 20^\circ} - \frac{1}{\cos 20^\circ}\)

Now, let's combine the two terms by finding a common denominator:

Expression = \(\frac{\sqrt3 \cos 20^\circ - 1 \cdot \sin 20^\circ}{\sin 20^\circ \cos 20^\circ}\)

Let's analyze the numerator: \(\sqrt3 \cos 20^\circ - \sin 20^\circ\). This looks similar to the expansion of sin(A - B) or cos(A + B). We can factor out 2 from the numerator to make it fit a standard trigonometric identity form:

Numerator = \(2 \left( \frac{\sqrt3}{2} \cos 20^\circ - \frac{1}{2} \sin 20^\circ \right)\)

We know that \(\frac{\sqrt3}{2} = \sin 60^\circ\) and \(\frac{1}{2} = \cos 60^\circ\). Substituting these values into the numerator:

Numerator = \(2 (\sin 60^\circ \cos 20^\circ - \cos 60^\circ \sin 20^\circ)\)

This expression inside the parentheses is the expansion of \(\sin(A - B)\) where A = 60° and B = 20°. The formula is \(\sin(A - B) = \sin A \cos B - \cos A \sin B\).

So, the numerator simplifies to:

Numerator = \(2 \sin (60^\circ - 20^\circ) = 2 \sin 40^\circ\)

Now, let's analyze the denominator: \(\sin 20^\circ \cos 20^\circ\). This is part of the double angle formula for sine: \(\sin 2\theta = 2 \sin \theta \cos \theta\).

We can rewrite the denominator using this identity:

Denominator = \(\frac{1}{2} (2 \sin 20^\circ \cos 20^\circ) = \frac{1}{2} \sin (2 \times 20^\circ) = \frac{1}{2} \sin 40^\circ\)

Now, substitute the simplified numerator and denominator back into the main expression:

Expression = \(\frac{2 \sin 40^\circ}{\frac{1}{2} \sin 40^\circ}\)

Assuming \(\sin 40^\circ \neq 0\) (which is true), we can cancel out \(\sin 40^\circ\) from the numerator and the denominator:

Expression = \(\frac{2}{\frac{1}{2}} = 2 \times 2 = 4\)

Thus, the value of \(\sqrt3\) cosec 20° - sec 20° is 4.

Key Trigonometric Identities Used

Revision Table: Important Values

Additional Information on Trigonometric Expressions

Evaluating trigonometric expressions often requires simplifying the expression using fundamental identities. These identities help transform complex expressions into simpler forms that can be evaluated directly or cancelled out.

Some common strategies for simplifying expressions include:

• Converting all trigonometric functions to sines and cosines.
• Finding a common denominator when adding or subtracting fractions.
• Using Pythagorean identities (e.g., \(\sin^2 \theta + \cos^2 \theta = 1\)).
• Applying sum/difference or double/half angle formulas.
• Factoring expressions.

Recognizing patterns that match parts of standard identities, like \(\sqrt3 \cos \theta - \sin \theta\) resembling parts of sum/difference formulas, is a key skill. Often, factoring out a constant (like 2 in this case) helps reveal the identity.