If `2sec^(2)theta+tan^(2)theta=17`, then find `cottheta=?`

Answer

To solve the equation \(2\sec^2\theta + \tan^2\theta = 17\) and find \(\cot\theta\), we can follow these steps:

### Step 1: Use the identity
We know the trigonometric identity:
\[ \sec^2\theta - \tan^2\theta = 1 \]
Let's call this Equation (1).

### Step 2: Express \(\sec^2\theta\) in terms of \(\tan^2\theta\)
From Equation (1), we can express \(\sec^2\theta\) as:
\[ \sec^2\theta = \tan^2\theta + 1 \]

### Step 3: Substitute \(\sec^2\theta\) into the original equation
Now, substitute \(\sec^2\theta\) into the original equation:
\[ 2(\tan^2\theta + 1) + \tan^2\theta = 17 \]

### Step 4: Simplify the equation
Expanding this gives:
\[ 2\tan^2\theta + 2 + \tan^2\theta = 17 \]
Combining like terms:
\[ 3\tan^2\theta + 2 = 17 \]

### Step 5: Solve for \(\tan^2\theta\)
Subtract 2 from both sides:
\[ 3\tan^2\theta = 15 \]
Now divide by 3:
\[ \tan^2\theta = 5 \]

### Step 6: Find \(\tan\theta\)
Taking the square root of both sides gives:
\[ \tan\theta = \sqrt{5} \]

### Step 7: Find \(\cot\theta\)
Recall that:
\[ \cot\theta = \frac{1}{\tan\theta} \]
So:
\[ \cot\theta = \frac{1}{\sqrt{5}} \]

### Final Answer
Thus, the value of \(\cot\theta\) is:
\[ \cot\theta = \frac{1}{\sqrt{5}} \]

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