The problem is to simplify the expression $M = (sin\theta - cos\theta)^2 + 2sin^2\theta + cot\theta$.

1. Problem Description

The problem is to simplify the expression $M = (sin\theta - cos\theta)^2 + 2sin^2\theta + cot\theta$.

2. Solution Steps

First, expand the square:

$(sin\theta - cos\theta)^2 = sin^2\theta - 2sin\theta cos\theta + cos^2\theta$

Substitute this back into the expression for M:

$M = sin^2\theta - 2sin\theta cos\theta + cos^2\theta + 2sin^2\theta + cot\theta$

Rearrange the terms:

$M = sin^2\theta + cos^2\theta + 2sin^2\theta - 2sin\theta cos\theta + cot\theta$

Use the identity $sin^2\theta + cos^2\theta = 1$:

$M = 1 + 2sin^2\theta - 2sin\theta cos\theta + cot\theta$

Rewrite $cot\theta$ as $\frac{cos\theta}{sin\theta}$:

$M = 1 + 2sin^2\theta - 2sin\theta cos\theta + \frac{cos\theta}{sin\theta}$

Combine the last two terms:

$M = 1 + 2sin^2\theta + \frac{cos\theta - 2sin^2\theta cos\theta}{sin\theta}$

Factor $cos\theta$ from the numerator:

$M = 1 + 2sin^2\theta + \frac{cos\theta (1 - 2sin^2\theta)}{sin\theta}$

Use the double angle formula $cos(2\theta) = 1 - 2sin^2\theta$:

$M = 1 + 2sin^2\theta + \frac{cos\theta cos(2\theta)}{sin\theta}$

$M = 1 + 2sin^2\theta + cot\theta cos(2\theta)$

However, let's retrace our steps from the earlier equation:

$M = 1 + 2sin^2\theta - 2sin\theta cos\theta + cot\theta$

$M = 1 + 2sin^2\theta - sin(2\theta) + cot\theta$

Also consider

$M = sin^2\theta - 2sin\theta cos\theta + cos^2\theta + 2sin^2\theta + \frac{cos\theta}{sin\theta}$

$M = 3sin^2\theta - 2sin\theta cos\theta + cos^2\theta + \frac{cos\theta}{sin\theta}$

$M = 3sin^2\theta - sin(2\theta) + cos^2\theta + \frac{cos\theta}{sin\theta}$

$M = 2sin^2\theta - sin(2\theta) + sin^2\theta + cos^2\theta + \frac{cos\theta}{sin\theta}$

$M = 2sin^2\theta - sin(2\theta) + 1 + \frac{cos\theta}{sin\theta}$

$M = 2sin^2\theta - sin(2\theta) + 1 + cot\theta$

$M = 1 - sin(2\theta) + cot\theta + 2sin^2\theta$

Let's go back to

$M = 1 + 2sin^2\theta - 2sin\theta cos\theta + cot\theta$

$M = 1 + 2sin^2\theta - sin2\theta + cot\theta$

Let's try $M = 1 + 2sin^2\theta - sin2\theta + \frac{cos\theta}{sin\theta} = 1 + 2sin^2\theta - sin2\theta + cot\theta$

3. Final Answer

$M = 1 + 2sin^2\theta - sin(2\theta) + cot\theta$