`sec(4x-50^(@))="cosec"(50^(@)-x)` then the value of x is

Options

A. `45^(@)`
B. `90^(@)`
C. `30^(@)`
D. `60^(@)`

Answer

To solve the equation \( \sec(4x - 50^\circ) = \csc(50^\circ - x) \), we will follow these steps:

### Step 1: Rewrite the Cosecant Function
We know that:
\[ \csc \theta = \sec(90^\circ - \theta) \]
Thus, we can rewrite the equation as:
\[ \sec(4x - 50^\circ) = \sec(90^\circ - (50^\circ - x)) \]

### Step 2: Simplify the Right Side
Now simplify the right side:
\[ 90^\circ - (50^\circ - x) = 90^\circ - 50^\circ + x = 40^\circ + x \]
So, we have:
\[ \sec(4x - 50^\circ) = \sec(40^\circ + x) \]

### Step 3: Set the Arguments Equal
Since the secant function is equal, we can set the arguments equal to each other:
\[ 4x - 50^\circ = 40^\circ + x \]

### Step 4: Solve for x
Now, let's solve for \( x \):
1. Rearrange the equation:
\[ 4x - x = 40^\circ + 50^\circ \]
\[ 3x = 90^\circ \]
2. Divide both sides by 3:
\[ x = \frac{90^\circ}{3} = 30^\circ \]

### Conclusion
The value of \( x \) is \( 30^\circ \).

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