Prove that (1 + $tan^2 A$) / (1 + $cot^2 A$) = $tan^2 A$.

Step 1: Recall Trigonometric Identities

We begin by recalling two fundamental trigonometric identities. The first identity states that $1 + \tan^2 A$ is equal to $\sec^2 A$. The second identity states that $1 + \cot^2 A$ is equal to $\csc^2 A$. 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 $\sec^2 A$ and $\csc^2 A$.

Step 3: Express in terms of Sine and Cosine

To further simplify, we express $\sec^2 A$ and $\csc^2 A$ in terms of their reciprocal functions, cosine and sine, respectively. We know that $\sec A = \frac{1}{\cos A}$ and $\csc A = \frac{1}{\sin A}$.

Step 4: Substitute Reciprocal Forms

Substitute the reciprocal forms of $\sec^2 A$ and $\csc^2 A$ 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 $\frac{\sin^2 A}{\cos^2 A}$.

Step 6: Final Simplification

Finally, we recognize that $\frac{\sin A}{\cos A}$ is equal to $\tan A$. Therefore, $\frac{\sin^2 A}{\cos^2 A}$ simplifies to $\tan^2 A$, which is the right-hand side of the given equation. This completes the proof.

More from this chapter

• **EXERCISE 8.3** **1.** Express the trigonometric ratios sin A, tan A and sec A in terms of cot A.
• Cos A / 1 + sin a + 1 + sin a / cos A is equal to 2 sec a
• write all the other trigonometric ratios of angle A in terms of sec A
• Express all the trigonometric ratios in terms of Sin A
• Express the trigonometric ratio sign a sec a tan a in terms of cot a
• Prove that √(sec² A + csc² A) = tan A + cot A
• (secθ - tanθ) = (1)/(m) --- (2) Adding (1) & (2) secθ + tanθ + secθ - tanθ = (1)/(m) + (m)/(1) 2secθ = (1 + m²)/(m) secθ = (1 + m²)/(2m) (Hence proved) 2) If as
• If tanθ + secθ = m, prove that secθ = (m² + 1)/(2m).

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