Trigonometric Expression Analysis

The problem requires simplifying the following trigonometric expression:

$ (\text{cosec}23^{\circ}-\text{sin}23^{\circ}) (\text{tan}23^{\circ} + \text{cot}23^{\circ}) (\text{sec}23^{\circ}-\text{cos}23^{\circ}) $

To solve this, we will simplify each part of the expression involving trigonometric functions like cosec, sin, tan, cot, sec, and cos, specifically at the angle of 23 degrees.

Simplifying Each Factor

We will rewrite the trigonometric functions in terms of sin and cos and simplify each parenthesis separately:

• First Factor: $(\text{cosec}23^{\circ}-\text{sin}23^{\circ})$
  • Recall that $\text{cosec}\theta = \frac{1}{\text{sin}\theta}$.
  • Substitute: $(\frac{1}{\text{sin}23^{\circ}}-\text{sin}23^{\circ})$
  • Find a common denominator: $\frac{1 - \text{sin}^223^{\circ}}{\text{sin}23^{\circ}}$
  • Using the Pythagorean identity $\text{sin}^2\theta + \text{cos}^2\theta = 1$, we know $1 - \text{sin}^2\theta = \text{cos}^2\theta$.
  • So, the first factor simplifies to: $\frac{\text{cos}^223^{\circ}}{\text{sin}23^{\circ}}$
• Second Factor: $(\text{tan}23^{\circ} + \text{cot}23^{\circ})$
  • Recall that $\text{tan}\theta = \frac{\text{sin}\theta}{\text{cos}\theta}$ and $\text{cot}\theta = \frac{\text{cos}\theta}{\text{sin}\theta}$.
  • Substitute: $(\frac{\text{sin}23^{\circ}}{\text{cos}23^{\circ}} + \frac{\text{cos}23^{\circ}}{\text{sin}23^{\circ}})$
  • Find a common denominator: $\frac{\text{sin}^223^{\circ} + \text{cos}^223^{\circ}}{\text{sin}23^{\circ}\text{cos}23^{\circ}}$
  • Using the identity $\text{sin}^2\theta + \text{cos}^2\theta = 1$.
  • So, the second factor simplifies to: $\frac{1}{\text{sin}23^{\circ}\text{cos}23^{\circ}}$
• Third Factor: $(\text{sec}23^{\circ}-\text{cos}23^{\circ})$
  • Recall that $\text{sec}\theta = \frac{1}{\text{cos}\theta}$.
  • Substitute: $(\frac{1}{\text{cos}23^{\circ}}-\text{cos}23^{\circ})$
  • Find a common denominator: $\frac{1 - \text{cos}^223^{\circ}}{\text{cos}23^{\circ}}$
  • Using the identity $1 - \text{cos}^2\theta = \text{sin}^2\theta$.
  • So, the third factor simplifies to: $\frac{\text{sin}^223^{\circ}}{\text{cos}23^{\circ}}$

Combining Factors for Solution

Now, multiply the simplified factors together:

Let's group the terms in the numerator and the denominator:

Now, form the fraction:

Final Calculation and Result

By canceling out the identical terms in the numerator and the denominator ($\text{cos}^223^{\circ}$ and $\text{sin}^223^{\circ}$), the expression simplifies to:

Therefore, the value of the given trigonometric expression is 1.