Step-by-Step Simplification

The expression \( A = \sin\theta \cos\theta \tan\theta + \sin\theta \cos\theta \cot\theta \) simplifies to 1.

Factor out the common term \( \sin\theta \cos\theta \) from both terms in the expression:

\[ A = \sin\theta \cos\theta (\tan\theta + \cot\theta)  \]

Substitute the definitions \( \tan\theta = \frac{\sin\theta}{\cos\theta} \) and \( \cot\theta = \frac{\cos\theta}{\sin\theta} \):

\[ \tan\theta + \cot\theta = \frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} = \frac{\sin^2\theta + \cos^2\theta}{\sin\theta \cos\theta} = \frac{1}{\sin\theta \cos\theta}

The expression becomes:

\[ A = \sin\theta \cos\theta \cdot \frac{1}{\sin\theta \cos\theta} = 1 \]

This holds for \( \theta \) where \( \sin\theta \neq 0 \) and \( \cos\theta \neq 0 \).

Options Analysis

Typical multiple-choice options for such CSIR NET-style questions include constants or trigonometric functions. All non-constant options fail:

• If an option is \( \sec\theta + \csc\theta \), it equals \( (\sin\theta + \cos\theta)(\tan\theta + \cot\theta) \), which differs by the \( \sin\theta \cos\theta \) factor. [web:18]
• If \( \tan\theta + \cot\theta \), this is \( \frac{1}{\sin\theta \cos\theta} \), larger than 1 by the same factor.
• If \( \sin 2\theta + \cos 2\theta \), it equals 1 but ignores the specific structure here.

The correct choice is always 1, verified algebraically and symbolically.

Why This Identity Matters

Simplifying expressions like \( \sin\theta \cos\theta (\tan\theta + \cot\theta) \) tests core identities in CSIR NET Life Sciences mathematics sections. The keyphrase “sinθ cosθ tanθ + sinθ cosθ cotθ value” reveals it equals 1 through factoring and Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \).

Such problems appear in competitive exams like CSIR NET, building skills in algebraic manipulation of trig functions. Recognizing \( \tan\theta + \cot\theta = \frac{1}{\sin\theta \cos\theta} \) cancels the leading factor instantly.

Common Exam Mistakes

• Forgetting to factor \( \sin\theta \cos\theta \).
• Misapplying reciprocal identities without common denominator.
• Confusing with similar identities like \( (\sin\theta + \cos\theta)(\tan\theta + \cot\theta) = \sec\theta + \csc\theta \).

Practice verifies the value remains 1 across quadrants where defined.