Understanding the Trigonometric Expression Problem

The question asks for the value of a given trigonometric expression involving $\sin A$, $\cos A$, $\tan A$, and $\cot A$. To find the value, we need to simplify the expression using basic trigonometric identities and algebraic formulas.

Step-by-Step Simplification Process

The given expression is:

\((1 + \cot A + \tan A)(\sin A - \cos A) \left(\frac{\sin A\cos A}{\sin^3A-\cos^3A}\right)\)

Simplifying the First Factor: \((1 + \cot A + \tan A)\)

We can express $\cot A$ and $\tan A$ in terms of $\sin A$ and $\cos A$:

\(\cot A = \frac{\cos A}{\sin A}\)

\(\tan A = \frac{\sin A}{\cos A}\)

Substitute these into the first factor:

\(1 + \cot A + \tan A = 1 + \frac{\cos A}{\sin A} + \frac{\sin A}{\cos A}\)

Find a common denominator, which is $\sin A \cos A$:

\(= \frac{\sin A \cos A}{\sin A \cos A} + \frac{\cos A \cdot \cos A}{\sin A \cos A} + \frac{\sin A \cdot \sin A}{\sin A \cos A}\)

\(= \frac{\sin A \cos A + \cos^2 A + \sin^2 A}{\sin A \cos A}\)

Using the fundamental trigonometric identity $\sin^2 A + \cos^2 A = 1$:

\(= \frac{\sin A \cos A + 1}{\sin A \cos A}\)

Simplifying the Denominator of the Third Factor: \((\sin^3A-\cos^3A)\)

This is a difference of cubes, which can be factored using the algebraic identity \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Here, \(a = \sin A\) and \(b = \cos A\).

\(\sin^3 A - \cos^3 A = (\sin A - \cos A)(\sin^2 A + \sin A \cos A + \cos^2 A)\)

Again using the identity $\sin^2 A + \cos^2 A = 1$:

\(= (\sin A - \cos A)(1 + \sin A \cos A)\)

Substituting Simplified Terms Back into the Expression

Now substitute the simplified forms back into the original expression:

\(\left(\frac{1 + \sin A \cos A}{\sin A \cos A}\right) (\sin A - \cos A) \left(\frac{\sin A\cos A}{(\sin A - \cos A)(1 + \sin A \cos A)}\right)\)

Let's rewrite the entire expression as a single fraction to see the terms clearly:

\(= \frac{(1 + \sin A \cos A)}{\sin A \cos A} \times (\sin A - \cos A) \times \frac{\sin A\cos A}{(\sin A - \cos A)(1 + \sin A \cos A)}\)

\(= \frac{(1 + \sin A \cos A) \times (\sin A - \cos A) \times (\sin A \cos A)}{(\sin A \cos A) \times (\sin A - \cos A) \times (1 + \sin A \cos A)}\)

Cancelling Common Factors

We can observe that there are common factors in the numerator and the denominator:

• \((1 + \sin A \cos A)\) appears in both numerator and denominator.
• \((\sin A - \cos A)\) appears in both numerator and denominator.
• \((\sin A \cos A)\) appears in both numerator and denominator.

Assuming $\sin A \ne 0$, $\cos A \ne 0$, $\sin A \ne \cos A$, and $1 + \sin A \cos A \ne 0$ (which are generally true unless A takes specific values that would make the original expression undefined anyway), we can cancel these terms:

\(= \frac{\cancel{(1 + \sin A \cos A)} \times \cancel{(\sin A - \cos A)} \times \cancel{(\sin A \cos A)}}{\cancel{(\sin A \cos A)} \times \cancel{(\sin A - \cos A)} \times \cancel{(1 + \sin A \cos A)}}\)

After cancellation, the expression simplifies to:

\(= 1\)

Conclusion on the Value of the Expression

The value of the given trigonometric expression \((1 + \cot A + \tan A)(\sin A - \cos A) \left(\frac{\sin A\cos A}{\sin^3A-\cos^3A}\right)\) simplifies to \(1\).

Revision Table: Key Trigonometric Identities Used

Additional Information on Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a common task in trigonometry. The general strategy involves:

• Converting all trigonometric functions to sine and cosine.
• Using fundamental identities like \(\sin^2 \theta + \cos^2 \theta = 1\), \(\sec^2 \theta - \tan^2 \theta = 1\), \(\csc^2 \theta - \cot^2 \theta = 1\).
• Applying algebraic formulas such as factoring (difference of squares, difference/sum of cubes), expanding, finding common denominators, etc.
• Cancelling common terms in the numerator and denominator.
• Recognizing and using sum/difference, double angle, half angle, or product-to-sum identities if applicable (though not needed in this specific problem).

It's crucial to remember the domain restrictions for functions like $\tan A$ and $\cot A$ and for expressions involving denominators to avoid division by zero.