sin 2a - (sin a + cos a)^2

Let's simplify the expression step by step:

Given: \[ \sin 2a - (\sin a + \cos a)^2 \]

Step 1: Recall the double angle formula: \[ \sin 2a = 2 \sin a \cos a \]

Step 2: Expand \((\sin a + \cos a)^2\): \[ (\sin a + \cos a)^2 = \sin^2 a + 2 \sin a \cos a + \cos^2 a \]

Step 3: Substitute the expansions into the original expression: \[ 2 \sin a \cos a - (\sin^2 a + 2 \sin a \cos a + \cos^2 a) \]

Step 4: Distribute the negative sign: \[ 2 \sin a \cos a - \sin^2 a - 2 \sin a \cos a - \cos^2 a \]

Step 5: Combine like terms:

• \(2 \sin a \cos a - 2 \sin a \cos a = 0\)
• Remaining terms: \(- \sin^2 a - \cos^2 a\)

Step 6: Recall the Pythagorean identity: \[ \sin^2 a + \cos^2 a = 1 \]

So, \(- \sin^2 a - \cos^2 a = - (\sin^2 a + \cos^2 a) = -1\)

Final answer: \[ \boxed{-1} \]

Therefore, \(\sin 2a - (\sin a + \cos a)^2 = -1\).