Solving Trigonometric Expression: Tan^2 Theta + Cot^2 Theta
We are given the equation:
$ \tan\theta + \cot\theta = 6 $
We need to find the value of the expression:
$ \tan^2\theta + \cot^2\theta $
Applying Algebraic Identity
Let's square both sides of the given equation:
$ (\tan\theta + \cot\theta)^2 = 6^2 $
Using the algebraic identity $ (a+b)^2 = a^2 + b^2 + 2ab $, we expand the left side:
$ \tan^2\theta + \cot^2\theta + 2(\tan\theta)(\cot\theta) = 36 $
Simplifying the Trigonometric Term
We know that $ \cot\theta = \frac{1}{\tan\theta} $. Therefore, the product $ (\tan\theta)(\cot\theta) $ simplifies to 1:
$ \tan\theta \cdot \cot\theta = \tan\theta \cdot \frac{1}{\tan\theta} = 1 $
Substituting this back into the expanded equation:
$ \tan^2\theta + \cot^2\theta + 2(1) = 36 $
$ \tan^2\theta + \cot^2\theta + 2 = 36 $
Calculating the Final Value
To find the required value, isolate $ \tan^2\theta + \cot^2\theta $ by subtracting 2 from both sides:
$ \tan^2\theta + \cot^2\theta = 36 - 2 $
$ \tan^2\theta + \cot^2\theta = 34 $
Thus, the value of $ \tan^2\theta + \cot^2\theta $ is 34.
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