Prove that (1 + tan2A) / (1 + cot2A) = tan2A.
Get the complete, step-by-step math solution for: "Prove that (1 + tan² A) / (1 + cot² A) = tan² A.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Recall Trigonometric Identities
We begin by recalling two fundamental trigonometric identities. The first identity states that 1+tan2A is equal to sec2A. The second identity states that 1+cot2A is equal to csc2A. These identities are crucial for simplifying the given expression.
Step 2: Substitute Identities into the Expression
Now, we substitute the identities from the previous step into the left-hand side of the given expression. This transforms the expression into a ratio of sec2A and csc2A.
Step 3: Express in terms of Sine and Cosine
To further simplify, we express sec2A and csc2A in terms of their reciprocal functions, cosine and sine, respectively. We know that secA=cosA1 and cscA=sinA1.
Step 4: Substitute Reciprocal Forms
Substitute the reciprocal forms of sec2A and csc2A into the expression. This gives us a complex fraction that can be simplified by multiplying by the reciprocal of the denominator.
Step 5: Simplify the Expression
To simplify the complex fraction, we multiply the numerator by the reciprocal of the denominator. This results in cos2Asin2A.
Step 6: Final Simplification
Finally, we recognize that cosAsinA is equal to tanA. Therefore, cos2Asin2A simplifies to tan2A, which is the right-hand side of the given equation. This completes the proof.