If `tan^(-1)(sin^2theta-2sintheta+3)+cot^(-1)(5^sec^(2y)+1)=pi/2,` then value of `cos^2theta-sintheta` is equal to 0 (b) `-1` (c) `1` (d) none of these

Options

A. 0
B. `-1`
C. 1
D. none of these

Explanation

To solve the equation

\[ \tan^{-1}(\sin^2 \theta - 2 \sin \theta + 3) + \cot^{-1}(5 \sec^2 y + 1) = \frac{\pi}{2}, \]

we can utilize the identity that states:

\[ \tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2}. \]

This means that we can set the arguments of the inverse functions equal to each other:

\[ \sin^2 \theta - 2 \sin \theta + 3 = 5 \sec^2 y + 1. \]

### Step 1: Simplify the left side

The left side can be rewritten as:

\[ \sin^2 \theta - 2 \sin \theta + 3 = (\sin \theta - 1)^2 + 2. \]

### Step 2: Set the equation

Now we can set the equation:

\[ (\sin \theta - 1)^2 + 2 = 5 \sec^2 y + 1. \]

### Step 3: Rearrange the equation

Rearranging gives us:

\[ (\sin \theta - 1)^2 + 1 = 5 \sec^2 y. \]

### Step 4: Analyze the range of \(\sin \theta\)

The term \((\sin \theta - 1)^2\) has a minimum value of 0 (when \(\sin \theta = 1\)) and a maximum value of 4 (when \(\sin \theta = -1\)). Therefore, the left side can range from:

\[ 1 \quad (\text{when } \sin \theta = 1) \quad \text{to} \quad 5 \quad (\text{when } \sin \theta = -1). \]

### Step 5: Analyze the right side

The right side is \(5 \sec^2 y\), which is always greater than or equal to 5 (since \(\sec^2 y \geq 1\)).

### Step 6: Set the ranges equal

For the equation to hold, we need:

\[ 5 \sec^2 y \geq 1 \quad \text{and} \quad 5 \sec^2 y \leq 5. \]

This implies that:

\[ \sec^2 y = 1 \quad \Rightarrow \quad y = 0. \]

### Step 7: Substitute back to find \(\sin \theta\)

Now substituting \(y = 0\) into the equation gives us:

\[ (\sin \theta - 1)^2 + 1 = 5 \cdot 1 \quad \Rightarrow \quad (\sin \theta - 1)^2 + 1 = 5. \]

### Step 8: Solve for \(\sin \theta\)

This simplifies to:

\[ (\sin \theta - 1)^2 = 4 \quad \Rightarrow \quad \sin \theta - 1 = \pm 2. \]

Thus, we have two cases:

1. \(\sin \theta - 1 = 2 \quad \Rightarrow \quad \sin \theta = 3\) (not possible since \(\sin \theta\) must be between -1 and 1).
2. \(\sin \theta - 1 = -2 \quad \Rightarrow \quad \sin \theta = -1\).

### Step 9: Calculate \(\cos^2 \theta - \sin \theta\)

If \(\sin \theta = -1\), then:

\[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - (-1)^2 = 0. \]

Now, substituting into \(\cos^2 \theta - \sin \theta\):

\[ \cos^2 \theta - \sin \theta = 0 - (-1) = 1. \]

### Final Answer

Thus, the value of \(\cos^2 \theta - \sin \theta\) is:

\[ \boxed{1}. \]