Prove the identities for sum-to-product in trigonometry.

The sum-to-product identities in trigonometry are used to simplify trigonometric expressions involving the sum or difference of two angles.

The first identity is:

sin(A+B) = sinAcosB + cosAsinB

To prove this identity, we start with the left-hand side:

sin(A+B) = sinAcosB + cosAsinB

= (sinAcosB + cosAsinB) / 1

= (sinAcosB + cosAsinB) / (cosAcosB + sinAsinB) * (cosAcosB + sinAsinB) / (cosAcosB + sinAsinB)

= (sinAcosB + cosAsinB)cosAcosB + (sinAcosB + cosAsinB)sinAsinB / (cosAcosB + sinAsinB)

= sinAcosAcosBcosB + sinAsinBcosAcosB + sinAcosAsinBsinA + cosAsinAsinBcosB / (cosAcosB + sinAsinB)

= sin(A+B) / (cosAcosB + sinAsinB)

Therefore, sin(A+B) = sinAcosB + cosAsinB.

The second identity is:

cos(A+B) = cosAcosB - sinAsinB

To prove this identity, we start with the left-hand side:

cos(A+B) = cosAcosB - sinAsinB

= (cosAcosB - sinAsinB) / 1

= (cosAcosB - sinAsinB) / (cosAcosB + sinAsinB) * (cosAcosB + sinAsinB) / (cosAcosB + sinAsinB)

= cosAcosBcosA - cosAcosBsinAsinB + sinAsinBcosA - sinAsinBsinB / (cosAcosB + sinAsinB)

= cosAcosBcosA - cosAcosBsinAsinB + sinAsinBcosA - sinAsinBsinB / (cosAcosB + sinAsinB)

= cos(A+B) / (cosAcosB + sinAsinB)

Therefore, cos(A+B) = cosAcosB - sinAsinB.

These identities can be used to