Question

Prove:
sin (pi/4 + x) = sqrt2/2 (cos x + sin x)

Answer 1

Apply the angle sum identity
`\sin(a + b) = \sin(a)\, \cos(b) + \cos(a)\, \sin(b)`.

Explanation:

By the angle sum identity,
`\sin(a + b) = \sin(a)\, \cos(b) + \cos(a)\, \sin(b)` for any angles
`a` and
`b`.

Apply this identity to rewrite the left-hand side of the equation:

`\begin{aligned}& \sin\left((\pi)/(4) + x\right) \\ =\; & \sin\left((\pi)/(4)\right)\, \cos(x) + \cos\left((\pi)/(4)\right)\, \sin(x) \end{aligned}`.

The angle
`(\pi/4)` is equivalent to
`45^(\circ)`, which corresponds to the isosceles right triangle:
`\sin(\pi/4) = √(2) / 2`,
`\cos(\pi/4) = √(2) / 2`.

Substitute these two values into the expression above and simplify:

`\begin{aligned}& \sin\left((\pi)/(4) + x\right) \\ =\; & \sin\left((\pi)/(4)\right)\, \cos(x) + \cos\left((\pi)/(4)\right)\, \sin(x) \\ =\; & (√(2))/(2)\, \cos(x) + (√(2))/(2)\, \sin(x) \\ =\; & (√(2))/(2) \, (\sin(x) + \cos(x))\end{aligned}`.

Thus,
`\sin((\pi/4) + x) = (√(2) / 2)\, (\sin(x) + \cos(x))` as requested.