By AAA similarity criterion, prove that (1 + tan^2 A) / (1 + cot^2 A) = tan^2 A.
Understand the Problem
The question involves proving a trigonometric identity. It uses similarity criteria and trigonometric identities to relate two expressions, aiming to show that they are equal.
Answer
$$ \frac{1 + \tan^2 A}{1 + \cot^2 A} = \tan^2 A $$
Answer for screen readers
The identity is proved: $$ \frac{1 + \tan^2 A}{1 + \cot^2 A} = \tan^2 A $$
Steps to Solve
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Set Up the Problem We start with the left-hand side of the identity we need to prove: $$ \frac{1 + \tan^2 A}{1 + \cot^2 A} $$
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Substitute Trigonometric Identities Recall that: $$ \tan A = \frac{\sin A}{\cos A} \quad \text{and} \quad \cot A = \frac{\cos A}{\sin A} $$
Now substitute these into the equation: $$ \frac{1 + \left(\frac{\sin^2 A}{\cos^2 A}\right)}{1 + \left(\frac{\cos^2 A}{\sin^2 A}\right)} $$
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Simplify the Numerator Combine the terms in the numerator: $$ 1 + \frac{\sin^2 A}{\cos^2 A} = \frac{\cos^2 A + \sin^2 A}{\cos^2 A} $$ Using the Pythagorean identity $\cos^2 A + \sin^2 A = 1$, we simplify it to: $$ = \frac{1}{\cos^2 A} $$
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Simplify the Denominator Similarly, simplify the denominator: $$ 1 + \frac{\cos^2 A}{\sin^2 A} = \frac{\sin^2 A + \cos^2 A}{\sin^2 A} $$ Again, using the Pythagorean identity, this simplifies to: $$ = \frac{1}{\sin^2 A} $$
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Combine the Simplified Expressions Now, we combine the simplified numerator and denominator: $$ \frac{\frac{1}{\cos^2 A}}{\frac{1}{\sin^2 A}} = \frac{\sin^2 A}{\cos^2 A} $$
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Final Step to Show Equality This final expression equals $\tan^2 A$: $$ \tan^2 A = \tan^2 A $$ Thus proving that: $$ \frac{1 + \tan^2 A}{1 + \cot^2 A} = \tan^2 A $$
The identity is proved: $$ \frac{1 + \tan^2 A}{1 + \cot^2 A} = \tan^2 A $$
More Information
This proof utilizes fundamental trigonometric identities and properties of right triangles, reinforcing the relationships among sine, cosine, and tangent. It highlights the importance of the Pythagorean identity in trigonometry.
Tips
- Misapplying the Pythagorean identity; ensure you combine $\sin^2 A + \cos^2 A = 1$ correctly.
- Failing to simplify the fractions completely before combining them; always check individual steps for simplification.
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