How to solve a quadratic inequality?

The sum-to-product identities in trigonometry can be proven using basic algebraic manipulations.

The first identity states that sin(a) + sin(b) = 2sin((a+b)/2)cos((a-b)/2). To prove this, we start with the right-hand side and use the double angle formula for sine and cosine:

2sin((a+b)/2)cos((a-b)/2) = 2sin((a/2)+(b/2))cos((a/2)-(b/2))
= (2sin((a/2))cos((a/2)))cos((b/2)) + (2cos((a/2))sin((b/2)))sin((a/2))
= sin(a)cos(b) + cos(a)sin(b)
= sin(a) + sin(b)

Therefore, the identity is proven.

The second identity states that cos(a) + cos(b) = 2cos((a+b)/2)cos((a-b)/2). To prove this, we start with the right-hand side and use the double angle formula for cosine:

2cos((a+b)/2)cos((a-b)/2) = 2cos((a/2)+(b/2))cos((a/2)-(b/2))
= (2cos((a/2))cos((b/2))) - (2sin((a/2))sin((b/2)))
= cos(a)cos(b) - sin(a)sin(b)
= cos(a) + cos(b)

Therefore, the identity is proven.

These identities are useful in simplifying trigonometric expressions and solving trigonometric equations.