If x = a sec θ cos Ø, y = b sec θ sin Ø and z = c tan θ, then find the value of (x²/a² + y²/b² - z²/c²).

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Given:

x = a sec θ cos Ø

y = b sec θ sin Ø

z = c tan θ

We need to find the value of (x²/a² + y²/b² - z²/c²).

Step 1: Calculate each term separately.

x²/a² = (a sec θ cos Ø)² / a² = (a² sec² θ cos² Ø) / a² = sec² θ cos² Ø

y²/b² = (b sec θ sin Ø)² / b² = (b² sec² θ sin² Ø) / b² = sec² θ sin² Ø

z²/c² = (c tan θ)² / c² = (c² tan² θ) / c² = tan² θ

Step 2: Substitute these into the expression:

x²/a² + y²/b² - z²/c² = sec² θ cos² Ø + sec² θ sin² Ø - tan² θ

Step 3: Factor sec² θ from the first two terms:

= sec² θ (cos² Ø + sin² Ø) - tan² θ

Since cos² Ø + sin² Ø = 1,

= sec² θ * 1 - tan² θ

= sec² θ - tan² θ

Step 4: Use the trigonometric identity:

sec² θ - tan² θ = 1

Therefore, the value of the expression is 1.

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