If sec x cosec x = 2, then what is the tan nx + cot nx equal to?
2
Finding the Value of tannx + cotnx Given sec x cosec x = 2
The problem asks us to find the value of the expression \(\tan^n x + \cot^n x\), given the condition \(\sec x \cosec x = 2\). We need to use the given equation to find the possible values of \(x\), and then evaluate the required expression.
Step 1: Simplify the Given Equation
We are given the equation:
\(\sec x \cosec x = 2\)
Recall the fundamental trigonometric identities:
- \(\sec x = \frac{1}{\cos x}\)
- \(\cosec x = \frac{1}{\sin x}\)
Substitute these into the given equation:
\(\left(\frac{1}{\cos x}\right) \left(\frac{1}{\sin x}\right) = 2\)
\(\frac{1}{\sin x \cos x} = 2\)
From this, we can find the value of \(\sin x \cos x\):
\(\sin x \cos x = \frac{1}{2}\)
Step 2: Use Double Angle Identity
We know the double angle identity for sine:
\(\sin 2x = 2 \sin x \cos x\)
Multiply the equation \(\sin x \cos x = \frac{1}{2}\) by 2 on both sides:
\(2 \sin x \cos x = 2 \times \frac{1}{2}\)
\(2 \sin x \cos x = 1\)
Using the identity, we get:
\(\sin 2x = 1\)
Step 3: Find the Value of x
The general solution for \(\sin \theta = 1\) is \(\theta = 2k\pi + \frac{\pi}{2}\), where \(k\) is an integer.
In our case, \(\theta = 2x\). So, we have:
\(2x = 2k\pi + \frac{\pi}{2}\)
Divide by 2 to find \(x\):
\(x = \frac{2k\pi}{2} + \frac{\pi}{4}\)
\(x = k\pi + \frac{\pi}{4}\)
This means \(x\) can be \(\frac{\pi}{4}, \pi + \frac{\pi}{4} = \frac{5\pi}{4}, 2\pi + \frac{\pi}{4} = \frac{9\pi}{4}\), and so on for \(k \ge 0\). It can also be \(-\pi + \frac{\pi}{4} = -\frac{3\pi}{4}\), etc., for \(k < 0\).
Step 4: Evaluate tan x and cot x for the Found Values of x
Let's find the values of \(\tan x\) and \(\cot x\) for \(x = k\pi + \frac{\pi}{4}\).
Recall the property that \(\tan(\theta + k\pi) = \tan \theta\) and \(\cot(\theta + k\pi) = \cot \theta\) for any integer \(k\).
So, \(\tan x = \tan\left(k\pi + \frac{\pi}{4}\right) = \tan\left(\frac{\pi}{4}\right)\).
We know that \(\tan\left(\frac{\pi}{4}\right) = 1\).
Similarly, \(\cot x = \cot\left(k\pi + \frac{\pi}{4}\right) = \cot\left(\frac{\pi}{4}\right)\).
We know that \(\cot\left(\frac{\pi}{4}\right) = 1\).
For any integer \(k\), if \(\sec x \cosec x = 2\), the value of \(x\) must be such that \(\tan x = 1\) and \(\cot x = 1\). Note that for \(x = k\pi + \pi/4\), \(\cos x\) and \(\sin x\) are non-zero, so \(\sec x\) and \(\cosec x\) are well-defined.
Step 5: Calculate tannx + cotnx
Now we need to find the value of \(\tan^n x + \cot^n x\).
Substitute the values \(\tan x = 1\) and \(\cot x = 1\) into the expression:
\(\tan^n x + \cot^n x = (1)^n + (1)^n\)
For any integer value of \(n\), \(1^n = 1\).
So, the expression becomes:
\(1 + 1 = 2\)
The value of \(\tan^n x + \cot^n x\) is always 2, regardless of the integer value of \(n\), provided that the condition \(\sec x \cosec x = 2\) holds.
The final answer is 2.
| Key Step | Mathematical Operation / Identity | Result |
|---|---|---|
| Given Equation | \(\sec x \cosec x = 2\) | |
| Substitute Identities | \(\sec x = 1/\cos x\), \(\cosec x = 1/\sin x\) | \(\frac{1}{\sin x \cos x} = 2\) |
| Rearrange | Isolate \(\sin x \cos x\) | \(\sin x \cos x = 1/2\) |
| Apply Double Angle Identity | \(\sin 2x = 2 \sin x \cos x\) | \(\sin 2x = 1\) |
| Solve for 2x | General solution for \(\sin \theta = 1\) | \(2x = 2k\pi + \pi/2\) |
| Solve for x | Divide by 2 | \(x = k\pi + \pi/4\) |
| Evaluate tan x | \(\tan(k\pi + \pi/4) = \tan(\pi/4)\) | \(\tan x = 1\) |
| Evaluate cot x | \(\cot(k\pi + \pi/4) = \cot(\pi/4)\) | \(\cot x = 1\) |
| Evaluate tannx + cotnx | Substitute values | \(1^n + 1^n = 1 + 1 = 2\) |
The calculation confirms that the expression \(\tan^n x + \cot^n x\) evaluates to 2 when \(\sec x \cosec x = 2\). This matches one of the given options.
Revision Table: Important Trigonometric Concepts
| Concept | Description | Formula/Identity |
|---|---|---|
| Reciprocal Identities | Define secant and cosecant in terms of cosine and sine. | \(\sec x = \frac{1}{\cos x}\), \(\cosec x = \frac{1}{\sin x}\) |
| Double Angle Identity for Sine | Relates sine of a double angle to sine and cosine of the angle. | \(\sin 2x = 2 \sin x \cos x\) |
| Periodicity of Tan and Cot | The value of tan and cot repeats every \(\pi\) radians. | \(\tan(\theta + k\pi) = \tan \theta\), \(\cot(\theta + k\pi) = \cot \theta\) (k is integer) |
| Specific Value \(\tan(\pi/4)\) | The tangent of 45 degrees (or \(\pi/4\) radians). | \(\tan(\pi/4) = 1\) |
| Specific Value \(\cot(\pi/4)\) | The cotangent of 45 degrees (or \(\pi/4\) radians). | \(\cot(\pi/4) = 1\) |
| General Solution for \(\sin \theta = 1\) | All possible angles \(\theta\) for which \(\sin \theta\) is 1. | \(\theta = 2k\pi + \frac{\pi}{2}\) (k is integer) |
Additional Information: Trigonometric Functions and Equations
This problem demonstrates how to solve a trigonometric equation and then use the resulting information to evaluate another expression. Key steps involve using fundamental identities to simplify the initial equation and recognizing standard forms like \(\sin \theta = 1\).
- The equation \(\sec x \cosec x = 2\) is equivalent to \(\sin x \cos x = 1/2\). This form is often easier to work with as it involves only sine and cosine.
- Multiplying by 2 helps connect \(\sin x \cos x\) to the double angle formula \(\sin 2x\). This is a common technique in solving trig equations.
- Solving \(\sin 2x = 1\) gives specific values for \(2x\). It's crucial to use the general solution \(2k\pi + \frac{\pi}{2}\) to capture all possibilities for \(x\).
- For the specific values of \(x\) derived (\(k\pi + \pi/4\)), both \(\tan x\) and \(\cot x\) always equal 1 due to the periodicity of these functions and their value at \(\pi/4\).
- The power 'n' in the final expression \(\tan^n x + \cot^n x\) becomes irrelevant because \(1^n\) is always 1 for any integer \(n\).
- Understanding the domains where \(\sec x\) and \(\cosec x\) are defined (where \(\cos x \ne 0\) and \(\sin x \ne 0\)) is important, though in this case, the solution \(x = k\pi + \pi/4\) naturally avoids these values.
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