If sin θ + cosec θ = 2, then the value of (sin$^{11}$ θ + cosec$^{11}$ θ) is :

Q: If sin θ + cosec θ = 2, then the value of (sin$^{11}$ θ + cosec$^{11}$ θ) is :

To solve the problem, we start with the given equation: \[ \sin \theta + \csc \theta = 2 \] We know that $\csc \theta = \frac{1}{\sin \theta}$. Substituting this into the equation, we have: \[ \sin \theta + \frac{1}{\sin \theta} = 2 \] Let $ x = \sin \theta $. Then the equation becomes: \[ x + \frac{1}{x} = 2 \] Multiplying through by $ x $ to eliminate the fraction, we get: \[ x^2 + 1 = 2x \] Rearranging gives: \[ x^2 - 2x + 1 = 0 \] This can be factored as: \[ (x - 1)^2 = 0 \] Thus, $ x = 1 $. Therefore, $\sin \theta = 1$. If $\sin \theta = 1$, then $\csc \theta = \frac{1}{\sin \theta} = 1$. Now, we need to find the value of $\sin^{11} \theta + \csc^{11} \theta$: \[ \sin^{11} \theta + \csc^{11} \theta = 1^{11} + 1^{11} = 1 + 1 = 2 \] Thus, the value of $\sin^{11} \theta + \csc^{11} \theta$ is 2, which corresponds to Option 3. **Explanation for why other options are incorrect:** 1. **Option 1 (3):** This option suggests that the sum is 3, which is incorrect because both $\sin^{11} \theta$ and $\csc^{11} \theta$ are 1, leading to a sum of 2, not 3. 2. **Option 2 (1):** This option suggests that the sum is 1, which is incorrect because each term $\sin^{11} \theta$ and $\csc^{11} \theta$ is 1, so their sum is 2, not 1. 3. **Option 4 ($2^{11}$):** This option suggests that the sum is $2^{11}$, which is incorrect because the actual sum is simply 2, not a power of 2. Therefore, Option 3 is the correct answer, as it accurately reflects the calculated value of $\sin^{11} \theta + \csc^{11} \theta$.

Question

Accepted Answer

To solve the problem, we start with the given equation:

$$ \sin \theta + \csc \theta = 2 $$

We know that $\csc \theta = \frac{1}{\sin \theta}$. Substituting this into the equation, we have:

$$ \sin \theta + \frac{1}{\sin \theta} = 2 $$

Let $ x = \sin \theta $. Then the equation becomes:

$$ x + \frac{1}{x} = 2 $$

Multiplying through by $ x $ to eliminate the fraction, we get:

$$ x^2 + 1 = 2x $$

Rearranging gives:

$$ x^2 - 2x + 1 = 0 $$

This can be factored as:

$$ (x - 1)^2 = 0 $$

Thus, $ x = 1 $. Therefore, $\sin \theta = 1$.

If $\sin \theta = 1$, then $\csc \theta = \frac{1}{\sin \theta} = 1$.

Now, we need to find the value of $\sin^{11} \theta + \csc^{11} \theta$:

$$ \sin^{11} \theta + \csc^{11} \theta = 1^{11} + 1^{11} = 1 + 1 = 2 $$

Thus, the value of $\sin^{11} \theta + \csc^{11} \theta$ is 2, which corresponds to Option 3.

**Explanation for why other options are incorrect:**

1. **Option 1 (3):** This option suggests that the sum is 3, which is incorrect because both $\sin^{11} \theta$ and $\csc^{11} \theta$ are 1, leading to a sum of 2, not 3.

2. **Option 2 (1):** This option suggests that the sum is 1, which is incorrect because each term $\sin^{11} \theta$ and $\csc^{11} \theta$ is 1, so their sum is 2, not 1.

3. **Option 4 ($2^{11}$):** This option suggests that the sum is $2^{11}$, which is incorrect because the actual sum is simply 2, not a power of 2.

Therefore, Option 3 is the correct answer, as it accurately reflects the calculated value of $\sin^{11} \theta + \csc^{11} \theta$.