Question

Let t = cosA + sinA
prove that t^2 = 1 + sin2A

Answer 1

`t= \cos A + \sin A`

`t^2=(\cos A + \sin A)^2 = \cos^2 A + 2 \cos A \sin A + \sin ^2 A`

`t^2= \cos^2 A+ \sin ^2 A + 2 \cos A \sin A`

`t^2= 1 + 2 \cos A \sin A \quad\checkmark`

`t^2= 1 + \sin 2A \quad\checkmark`

We use the Pythagorean Theorem
`\cos^2 A + \sin ^2 A = 1` and the double angle formula for sine,
`\sin 2A = 2 \sin A \cos A`

Answer 2