If cosecθ=secθ, then value of θ is

Answer

To solve the equation \( \csc \theta = \sec \theta \), we can follow these steps:

### Step 1: Write the definitions of cosecant and secant
The cosecant and secant functions are defined as:
\[ \csc \theta = \frac{1}{\sin \theta} \quad \text{and} \quad \sec \theta = \frac{1}{\cos \theta} \]

### Step 2: Set up the equation
From the given equation \( \csc \theta = \sec \theta \), we can substitute the definitions:
\[ \frac{1}{\sin \theta} = \frac{1}{\cos \theta} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ \cos \theta = \sin \theta \]

### Step 4: Solve the equation
The equation \( \cos \theta = \sin \theta \) implies that:
\[ \tan \theta = 1 \]
This occurs when \( \theta = 45^\circ + n \cdot 180^\circ \) where \( n \) is any integer.

### Step 5: Determine the principal value
The principal value of \( \theta \) that satisfies this equation in the range of \( 0^\circ \) to \( 360^\circ \) is:
\[ \theta = 45^\circ \]

### Final Answer
Thus, the value of \( \theta \) is:
\[ \theta = 45^\circ \]

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