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.
Similar Questions
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Important Questions from Trigonometric Ratios and Identities
What is the ratio of the greatest to the smallest value of 2 – 2 sin x – sin 2x, 0 ≤ x ≤ (π/2)?
If sinθ = \(\frac{4}{5}\) , Find the value of sin3θ
If x, y are acute angles, where 0 < x + y < 90° and sin(3x - 40°) = cos (3y + 40°), then the value of tan (x + y) is equal to
What is (1 + cot θ - cosec θ)(1 + tan θ + sec θ) equal to?
If \(\sin \theta =\frac{3}{5}\) and \(\cos \theta =\frac{4}{5}\) , then the value of \(\frac{1+\tan \theta}{1-\cot \theta}\) is:
