If 3‾√ cot2θ – 4 cot θ + 3‾√ = 0, then find the value of cot2 θ + tan2θ. (2013)
Solution:
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Step-by-Step Solution
Step 1: Solve the quadratic equation for cot θ
We are given a quadratic equation in terms of cotθ. We can solve this quadratic equation for cotθ using the quadratic formula, where a=3, b=−4, and c=3.
Step 2: Apply the quadratic formula
Substitute the values of a, b, and c into the quadratic formula: x=2a−b±b2−4ac. This will give us the possible values for cotθ.
Step 3: Simplify the expression for cot θ
Simplify the expression under the square root and then calculate the two possible values for cotθ. This leads to two distinct solutions.
Step 4: Calculate the values of cot θ
Further simplify the two values of cotθ. We rationalize the denominators by multiplying the numerator and denominator by 3 where necessary.
Step 5: Calculate cot²θ and tan²θ for each case
For each value of cotθ, we calculate cot2θ and then find tanθ using the identity tanθ=cotθ1, and subsequently tan2θ.
Step 6: Find the value of cot²θ + tan²θ
Finally, add the calculated values of cot2θ and tan2θ. Both cases yield the same result.