Question

Prove the identity sin a + cos a = √2 sin (a + 45°)

Answer 1

To prove the trigonometric identity sin a + cos a = √2 sin (a + 45°), we use the sum of angles formula for sine and the values for cos 45° and sin 45°. Through substitution and simplification, the identity is shown to be true.

Step-by-step explanation:

To prove the identity sin a + cos a = √2 sin (a + 45°), we can use the sum of angles formula for sine. The sum of angles formula states sin (a ± β) = sin a cos β ± cos a sin β. We want to express the left side in the form of this formula, so we look at the right side:

sin (a + 45°) = sin a cos 45° + cos a sin 45°

Since cos 45° and sin 45° are both √2/2, this simplifies to:

sin a (√2/2) + cos a (√2/2) = (√2/2)(sin a + cos a)

Multiplying both sides by √2, we get:

√2 sin a + √2 cos a = √2 (√2/2)(sin a + cos a)

Now simplify by cancelling out the √2:

sin a + cos a = (√2 × √2/2)(sin a + cos a)

Thus:

sin a + cos a = √2 sin (a + 45°)

This completes the proof of the given trigonometric identity.