If 3‾√ cot2θ – 4 cot θ + 3‾√ = 0, then find the value of cot2 θ + tan2θ. (2013) Solution:

Step 1: Solve the quadratic equation for cot θ

We are given a quadratic equation in terms of $\cot\theta$. We can solve this quadratic equation for $\cot\theta$ using the quadratic formula, where $a = \sqrt{3}$, $b = -4$, and $c = \sqrt{3}$.

Step 2: Apply the quadratic formula

Substitute the values of $a$, $b$, and $c$ into the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This will give us the possible values for $\cot\theta$.

Step 3: Simplify the expression for cot θ

Simplify the expression under the square root and then calculate the two possible values for $\cot\theta$. This leads to two distinct solutions.

Step 4: Calculate the values of cot θ

Further simplify the two values of $\cot\theta$. We rationalize the denominators by multiplying the numerator and denominator by $\sqrt{3}$ where necessary.

Step 5: Calculate cot²θ and tan²θ for each case

For each value of $\cot\theta$, we calculate $\cot^2\theta$ and then find $\tan\theta$ using the identity $\tan\theta = \frac{1}{\cot\theta}$, and subsequently $\tan^2\theta$.

Step 6: Find the value of cot²θ + tan²θ

Finally, add the calculated values of $\cot^2\theta$ and $\tan^2\theta$. Both cases yield the same result.

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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