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