Given:
cot(theta) = 1/√2

To prove:
1 - cos²(theta) / 2 - sin²(theta) = 3/4

Proof:

Step 1: Find the values of sin(theta) and cos(theta) using the given cot(theta) value.

We know that cot(theta) = 1/√2
We also know that cot(theta) = cos(theta) / sin(theta)

So, we can write the equation as:
cos(theta) / sin(theta) = 1/√2

Cross multiplying, we get:
cos(theta) = sin(theta) / √2

Squaring both sides, we get:
cos²(theta) = sin²(theta) / 2

Step 2: Substitute the values of cos²(theta) and sin²(theta) in the expression to be proved.

1 - cos²(theta) / 2 - sin²(theta) = 3/4

Substituting the values:
1 - (sin²(theta) / 2) / 2 - sin²(theta) = 3/4

Step 3: Simplify the expression.

1 - (sin²(theta) / 2) = 3/4 + sin²(theta)

Multiplying all terms by 4, we get:
4 - 2sin²(theta) = 3 + 4sin²(theta)

Combining like terms, we get:
4 - 3 = 4sin²(theta) + 2sin²(theta)

1 = 6sin²(theta)

Dividing both sides by 6, we get:
1/6 = sin²(theta)

Step 4: Substitute the value of sin²(theta) back into the expression to be proved.

1 - cos²(theta) / 2 - sin²(theta) = 3/4

Substituting sin²(theta) = 1/6:
1 - cos²(theta) / 2 - 1/6 = 3/4

Multiplying all terms by 12 to eliminate fractions, we get:
12 - 6cos²(theta) - 2 = 9

Combining like terms, we get:
-6cos²(theta) + 10 = 9

Rearranging the equation, we get:
-6cos²(theta) = -1

Dividing both sides by -6, we get:
cos²(theta) = 1/6

Step 5: Verify that the value of cos²(theta) satisfies the equation.

1 - cos²(theta) / 2 - sin²(theta) = 3/4

Substituting cos²(theta) = 1/6 and sin²(theta) = 1/6:
1 - 1/6 / 2 - 1/6 = 3/4

Simplifying the expression, we get:
1 - 1/12 - 1/6 = 3/4

Combining like terms, we get:
(12 - 1 - 2)/12 = 3/4

9/12 = 3/4

Simplifying further, we get:
3/4 = 3/4

Therefore, the equation 1 - cos

Question change 🙄