If `tanx+secx=2cot(90^(@)+x)`, then `cosecx=?`

Options

A. `-1`
B. `-3`
C. `-5`
D. `-9`

Answer

To solve the equation \( \tan x + \sec x = 2 \cot(90^\circ + x) \) and find the value of \( \csc x \), we can follow these steps:

### Step 1: Simplify \( \cot(90^\circ + x) \)
Using the trigonometric identity, we know that:
\[ \cot(90^\circ + x) = -\tan x \]
So, we can rewrite the equation as:
\[ \tan x + \sec x = 2(-\tan x) \]
This simplifies to:
\[ \tan x + \sec x = -2\tan x \]

### Step 2: Rearranging the equation
Now, let's move \( \tan x \) to the right side:
\[ \tan x + 2\tan x + \sec x = 0 \]
This simplifies to:
\[ 3\tan x + \sec x = 0 \]

### Step 3: Express \( \sec x \) in terms of \( \tan x \)
We know that:
\[ \sec x = \frac{1}{\cos x} \quad \text{and} \quad \tan x = \frac{\sin x}{\cos x} \]
Substituting \( \sec x \) into the equation gives us:
\[ 3\tan x = -\sec x \]
Substituting for \( \sec x \):
\[ 3\frac{\sin x}{\cos x} = -\frac{1}{\cos x} \]

### Step 4: Multiply through by \( \cos x \) (assuming \( \cos x \neq 0 \))
This leads to:
\[ 3\sin x = -1 \]
So we can express \( \sin x \) as:
\[ \sin x = -\frac{1}{3} \]

### Step 5: Find \( \csc x \)
The cosecant function is the reciprocal of the sine function:
\[ \csc x = \frac{1}{\sin x} \]
Substituting the value of \( \sin x \):
\[ \csc x = \frac{1}{-\frac{1}{3}} = -3 \]

### Final Answer
Thus, the value of \( \csc x \) is:
\[ \csc x = -3 \]

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