Trigonometric Simplification: Analyzing the Expression

This solution provides a detailed, step-by-step breakdown for simplifying the complex trigonometric expression $\frac{\sin A - 2\sin^3 A}{2\cos^3 A - \cos A} + \sqrt{\text{cosec}^2 A - 1}$.

Simplifying the Fraction Term

The first part of the expression is the fraction: $ \frac{\sin A - 2\sin^3 A}{2\cos^3 A - \cos A} $ Let's simplify this using algebraic factoring and trigonometric identities.

• Numerator Simplification: Factor out the common term $\sin A$. $ \sin A (1 - 2\sin^2 A) $ We recognize $1 - 2\sin^2 A$ as the double angle identity for cosine, $\cos(2A)$. So the numerator is: $ \sin A \cos(2A) $
• Denominator Simplification: Factor out the common term $\cos A$. $ \cos A (2\cos^2 A - 1) $ We recognize $2\cos^2 A - 1$ as another form of the double angle identity for cosine, $\cos(2A)$. So the denominator is: $ \cos A \cos(2A) $

Substituting these back into the fraction gives:

$ \frac{\sin A \cos(2A)}{\cos A \cos(2A)} $

Assuming $\cos(2A) \neq 0$, we cancel out the $\cos(2A)$ term. This leaves us with:

$ \frac{\sin A}{\cos A} $

Which simplifies to $\tan A$.

Simplifying the Square Root Term

The second part of the expression is $\sqrt{\text{cosec}^2 A - 1}$.

We use the Pythagorean identity: $1 + \cot^2 A = \text{cosec}^2 A$. Rearranging this gives $\text{cosec}^2 A - 1 = \cot^2 A$.

Substituting this into the square root:

$ \sqrt{\cot^2 A} $

This simplifies to $|\cot A|$. For typical trigonometric simplification problems where the final answer is one of the standard options, we often assume the angle $A$ is such that $\cot A$ is positive, so we use $\cot A$.

Combining Simplified Terms

Now, we add the simplified terms from the previous steps:

$ \tan A + \cot A $

To combine these, convert them into $\sin A$ and $\cos A$:

$ \frac{\sin A}{\cos A} + \frac{\cos A}{\sin A} $

Find a common denominator ($\cos A \sin A$):

$ \frac{\sin^2 A}{\cos A \sin A} + \frac{\cos^2 A}{\cos A \sin A} $ $ \frac{\sin^2 A + \cos^2 A}{\cos A \sin A} $

Apply the main Pythagorean identity $\sin^2 A + \cos^2 A = 1$:

$ \frac{1}{\cos A \sin A} $

Using the reciprocal identities $\sec A = 1 / \cos A$ and $\text{cosec} A = 1 / \sin A$, we get:

$ \sec A \cdot \text{cosec} A $

Final Simplified Value

The expression simplifies completely to $\boldsymbol{\sec A \text{ cosec} A}$.