The value of $\frac{1 + \tan^2 \theta}{1 + \cot^2 \theta}$ is :

Step 1: Apply trigonometric identities

We use the fundamental trigonometric identities: $1 + \tan^2 \theta = \sec^2 \theta$ and $1 + \cot^2 \theta = \csc^2 \theta$. Substituting these into the given expression simplifies it.

Step 2: Express in terms of sine and cosine

Next, we express $\sec^2 \theta$ and $\csc^2 \theta$ in terms of $\sin \theta$ and $\cos \theta$. We know that $\sec \theta = \frac{1}{\cos \theta}$ and $\csc \theta = \frac{1}{\sin \theta}$.

Step 3: Simplify the complex fraction

To simplify the complex fraction, we multiply the numerator by the reciprocal of the denominator. This results in $\frac{\sin^2 \theta}{\cos^2 \theta}$.

Step 4: Convert to tangent

Finally, we recognize that $\frac{\sin \theta}{\cos \theta} = \tan \theta$. Therefore, the expression simplifies to $\tan^2 \theta$.

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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