Simplifying Trigonometric Expressions

The problem asks us to find the value of the given trigonometric expression: $\small (\tan^2 A + \cot^2 A - 2) - \sec^2 A \operatorname{cosec}^2 A$. To solve this, we will use fundamental trigonometric identities to simplify the expression step-by-step.

Step 1: Simplify the First Part of the Expression

Consider the first part of the expression: $\small \tan^2 A + \cot^2 A - 2$. We can use the identities:

• $\tan^2 A = \sec^2 A - 1$
• $\cot^2 A = \operatorname{cosec}^2 A - 1$

Substitute these identities into the first part:

$\small \tan^2 A + \cot^2 A - 2 = (\sec^2 A - 1) + (\operatorname{cosec}^2 A - 1) - 2$

Combine the terms:

$\small = \sec^2 A + \operatorname{cosec}^2 A - 1 - 1 - 2$

$\small = \sec^2 A + \operatorname{cosec}^2 A - 4$

Step 2: Rewrite the Second Part using Reciprocal Identities

Now consider the second part of the original expression: $\small \sec^2 A \operatorname{cosec}^2 A$. We use the reciprocal identities:

• $\sec A = \frac{1}{\cos A}$, so $\sec^2 A = \frac{1}{\cos^2 A}$
• $\operatorname{cosec} A = \frac{1}{\sin A}$, so $\operatorname{cosec}^2 A = \frac{1}{\sin^2 A}$

Substitute these into the second part:

$\small \sec^2 A \operatorname{cosec}^2 A = \left(\frac{1}{\cos^2 A}\right) \left(\frac{1}{\sin^2 A}\right) = \frac{1}{\cos^2 A \sin^2 A}$

Step 3: Substitute Simplified Parts back into the Original Expression

The original expression is $\small (\tan^2 A + \cot^2 A - 2) - \sec^2 A \operatorname{cosec}^2 A$. Substitute the simplified forms from Step 1 and Step 2:

$\small = (\sec^2 A + \operatorname{cosec}^2 A - 4) - \frac{1}{\cos^2 A \sin^2 A}$

Now, rewrite $\sec^2 A$ and $\operatorname{cosec}^2 A$ in terms of $\sin A$ and $\cos A$ again:

$\small = \left(\frac{1}{\cos^2 A} + \frac{1}{\sin^2 A} - 4\right) - \frac{1}{\cos^2 A \sin^2 A}$

Step 4: Combine and Simplify Further

Find a common denominator for the terms inside the first parenthesis:

$\small \frac{1}{\cos^2 A} + \frac{1}{\sin^2 A} = \frac{\sin^2 A}{\cos^2 A \sin^2 A} + \frac{\cos^2 A}{\cos^2 A \sin^2 A} = \frac{\sin^2 A + \cos^2 A}{\cos^2 A \sin^2 A}$

Using the Pythagorean identity $\small \sin^2 A + \cos^2 A = 1$, we get:

$\small \frac{\sin^2 A + \cos^2 A}{\cos^2 A \sin^2 A} = \frac{1}{\cos^2 A \sin^2 A}$

Substitute this back into the expression from Step 3:

$\small = \left(\frac{1}{\cos^2 A \sin^2 A} - 4\right) - \frac{1}{\cos^2 A \sin^2 A}$

Rearrange the terms:

$\small = \frac{1}{\cos^2 A \sin^2 A} - \frac{1}{\cos^2 A \sin^2 A} - 4$

The terms $\small \frac{1}{\cos^2 A \sin^2 A}$ and $\small - \frac{1}{\cos^2 A \sin^2 A}$ cancel each other out:

$\small = 0 - 4$

$\small = -4$

Result

The value of the expression $\small (\tan 2 A + \cot 2 A - 2) - \sec 2 A \operatorname{cosec} 2 A$ is $\small -4$.

Revision Table: Key Trigonometric Identities

Additional Information: Why Identities are Useful

Trigonometric identities are equations involving trigonometric functions that are true for every single value of the variables for which both sides of the equation are defined. They are crucial tools in trigonometry and calculus because they allow us to simplify complex expressions, solve trigonometric equations, and evaluate integrals. By transforming expressions into simpler forms using identities, calculations become much easier. In this problem, using identities like $\small \tan^2 A = \sec^2 A - 1$ and $\small \cot^2 A = \operatorname{cosec}^2 A - 1$ was key to simplifying the expression before combining terms with reciprocal identities and the Pythagorean identity.