Proving the Trigonometric Identity: (tan^2 A - 1) / (tan^2 A 1) sin^2 A - cos^2 A
In this article, we will demonstrate the proof of the trigonometric identity:
(frac{tan^2 A - 1}{tan^2 A 1} sin^2 A - cos^2 A)
First, let's express the tangent in terms of sine and cosine:
Step 1: Express tan A in terms of sin A and cos A
The tangent of angle A can be expressed as:
(tan A frac{sin A}{cos A})
Therefore, when squared, it becomes:
(tan^2 A frac{sin^2 A}{cos^2 A})
Now, we substitute (tan^2 A) into the left-hand side of the identity:
Step 2: Substitute tan^2 A into the left-hand side
(frac{tan^2 A - 1}{tan^2 A 1} frac{frac{sin^2 A}{cos^2 A} - 1}{frac{sin^2 A}{cos^2 A} 1})
To simplify, we can rewrite (1) in terms of (cos^2 A):(1 frac{cos^2 A}{cos^2 A})
Step 3: Rewrite the expression
Substituting this into the equation:
(frac{frac{sin^2 A - cos^2 A}{cos^2 A}}{frac{sin^2 A cos^2 A}{cos^2 A}} frac{sin^2 A - cos^2 A}{sin^2 A cos^2 A})
Next, we use the Pythagorean identity (sin^2 A cos^2 A 1) to simplify the denominator:
(frac{sin^2 A - cos^2 A}{1} sin^2 A - cos^2 A)
Therefore, we have shown that:
(frac{tan^2 A - 1}{tan^2 A 1} sin^2 A - cos^2 A)
This completes the proof of the identity.
Additional Proofs for Trigonometric Identity
Proof 2: Simplified Approach
Let us put everything in terms of sine and cosine. Starting with the identity:
(tan^2 A - 1 / tan^2 A 1)
Knowing that (tan A sin A / cos A), we can write:
(frac{tan^2 A - 1}{tan^2 A 1} frac{frac{sin^2 A}{cos^2 A} - 1}{frac{sin^2 A}{cos^2 A} 1})
Multiply both the numerator and the denominator by (cos^2 A):
(frac{sin^2 A - cos^2 A}{sin^2 A cos^2 A})
Since (sin^2 A cos^2 A 1), we can simplify the denominator:
(frac{sin^2 A - cos^2 A}{1} sin^2 A - cos^2 A)
Thus, we have proved:
(frac{tan^2 A - 1}{tan^2 A 1} sin^2 A - cos^2 A)
Conclusion
This identity is a demonstration of the interconnectedness of trigonometric functions. Understanding and proving such identities is crucial in trigonometry and helps in solving more complex problems involving trigonometric functions.