If sin theta - cos theta = 1 and theta is an acute angle, we need to find the value of 2cot² theta - sin² theta.

To solve this problem, we will use trigonometric identities and manipulate the given equation to find the value of the expression.

Step 1: Manipulating the given equation

We are given sin theta - cos theta = 1. We can rewrite this equation using the Pythagorean identity for sine and cosine:

sin theta - cos theta = sin theta - (1 - sin² theta)^(1/2) = 1

Simplifying further, we get:

sin theta - 1 + sin² theta = sin theta + sin² theta = 1

Step 2: Using the Pythagorean identity

Since theta is an acute angle, we can use the Pythagorean identity to relate sine and cosine:

sin² theta + cos² theta = 1

Rearranging, we have:

cos² theta = 1 - sin² theta

Step 3: Substituting the values in the expression

Now, we can substitute the value of cos² theta from Step 2 into the expression 2cot² theta - sin² theta:

2cot² theta - sin² theta = 2(cosec² theta - 1) - sin² theta

Using the reciprocal identity for cotangent, we have:

2(cosec² theta - 1) - sin² theta = 2(1 + cot² theta - 1) - sin² theta

Simplifying, we get:

2cot² theta - sin² theta = 2cot² theta - sin² theta

Step 4: Simplifying the expression

Since the expression in the question is equal to 1, we can write:

2cot² theta - sin² theta = 1

Conclusion:

After manipulating the given equation and substituting values, we found that the expression 2cot² theta - sin² theta is equal to 1.