Trigonometric Expression Simplification

Expression Variables Defined

We are given the following relationships involving trigonometric functions and variables:

• $x = a \text{ cosec}A \text{ sin} B$
• $y = b \text{ cosec}A \text{ cos} B$
• $z = c \text{ cot} A$

Expression Value Calculation

Our goal is to determine the value of the expression $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2}$.

x, y, z Term Calculation

Let's first calculate the individual squared terms required for the expression. We rearrange each given equation to isolate the ratio of the variable to its corresponding constant, and then square it.

• From $x = a \text{ cosec}A \text{ sin} B$, dividing by $a$ gives $\frac{x}{a} = \text{cosec}A \text{ sin} B$. Squaring both sides yields $\frac{x^2}{a^2} = (\text{cosec}A \text{ sin} B)^2 = \text{cosec}^2A \text{ sin}^2B$.
• From $y = b \text{ cosec}A \text{ cos} B$, dividing by $b$ gives $\frac{y}{b} = \text{cosec}A \text{ cos} B$. Squaring both sides yields $\frac{y^2}{b^2} = (\text{cosec}A \text{ cos} B)^2 = \text{cosec}^2A \text{ cos}^2B$.
• From $z = c \text{ cot} A$, dividing by $c$ gives $\frac{z}{c} = \text{cot} A$. Squaring both sides yields $\frac{z^2}{c^2} = (\text{cot} A)^2 = \text{cot}^2A$.

Expression Substitution

Now, we substitute these derived squared terms into the target expression:

Expression = $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2}$

Substituting the results from the previous step:

Expression = $(\text{cosec}^2A \text{ sin}^2B) + (\text{cosec}^2A \text{ cos}^2B) - (\text{cot}^2A)$

Simplification Using Trigonometric Identities

We can simplify this expression by factoring and applying fundamental trigonometric identities.

Notice that $\text{cosec}^2A$ is a common factor in the first two terms:

Expression = $\text{cosec}^2A (\text{sin}^2B + \text{cos}^2B) - \text{cot}^2A$

Using the Pythagorean identity $\text{sin}^2\theta + \text{cos}^2\theta = 1$, we know that $\text{sin}^2B + \text{cos}^2B = 1$. Substituting this:

Expression = $\text{cosec}^2A (1) - \text{cot}^2A$

This simplifies to:

Expression = $\text{cosec}^2A - \text{cot}^2A$

Again, applying a Pythagorean identity, $\text{cosec}^2\theta - \text{cot}^2\theta = 1$. Therefore:

Expression = $1$

Conclusion of Calculation

The value of the given trigonometric expression $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2}$ simplifies to 1.