Solving tan θ + cot θ = 2 for tan^53 θ + cot^38 θ
This problem requires us to first solve a basic trigonometric equation and then use the result to evaluate a more complex expression involving powers of tangent and cotangent.
Given Trigonometric Equation
We are provided with the following equation:
$$ \tan \theta + \cot \theta = 2 $$Our goal is to find the value of:
$$ \tan^{53} \theta + \cot^{38} \theta $$Solving Steps
Simplify tan θ + cot θ = 2
We begin by simplifying the given equation. Recall the identity $\cot \theta = \frac{1}{\tan \theta}$. Substituting this into the equation gives:
$$ \tan \theta + \frac{1}{\tan \theta} = 2 $$Let's represent $\tan \theta$ with a variable, say $x$. So, $x = \tan \theta$. The equation now looks like:
$$ x + \frac{1}{x} = 2 $$Solve for tan θ Value
To solve for $x$, we can clear the fraction by multiplying both sides by $x$. We assume $\tan \theta \neq 0$ because if it were, $\cot \theta$ would be undefined.
$$ x \left( x + \frac{1}{x} \right) = 2x $$ $$ x^2 + 1 = 2x $$Rearranging this into a standard quadratic form:
$$ x^2 - 2x + 1 = 0 $$This quadratic equation is a perfect square: $(x - 1)^2 = 0$. Taking the square root of both sides gives:
$$ x - 1 = 0 $$ $$ x = 1 $$Since we defined $x = \tan \theta$, we have:
$$ \tan \theta = 1 $$Find cot θ Value
Now that we know $\tan \theta = 1$, we can easily find $\cot \theta$ using the reciprocal identity:
$$ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{1} $$ $$ \cot \theta = 1 $$These values satisfy the original condition $\tan \theta + \cot \theta = 1 + 1 = 2$. This confirms our intermediate results.
Evaluate Expression tan^53 θ + cot^38 θ
We need to calculate the value of $\tan^{53} \theta + \cot^{38} \theta$. Substitute the values we found:
$$ \tan^{53} \theta + \cot^{38} \theta = (1)^{53} + (1)^{38} $$Calculate Final Value
Any positive integer power of 1 is 1:
$$ 1^{53} = 1 $$ $$ 1^{38} = 1 $$Adding these results together:
$$ 1 + 1 = 2 $$Thus, the value of the expression is 2.
Final Answer
The value of $\tan^{53} \theta + \cot^{38} \theta$ is 2.
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