# PreCalc 12 Help, Identities?

The question is a proof: sin(A-B)cosB+cos(A-B)sinB = sinA.. If someone could write out the left side and show me how using the identities you can change it into sinA it would help a lot. Thanks :)

### 2 Answers

- GridLv 78 years agoFavorite Answer
This one is true. Your previous example was not, you might want to re-look that one.

working on the LHS:

sin(A-B)cosB + cos(A-B)sinB

use following formulas:

sin(A-B) = sin(A)cos(B) - cos(A)sin(B)

cos(A - B) = sin(A)sin(B) + cos(A)cos(B)

[sin(A)cos(B) - cos(A)sin(B)] * cos(B) + [sin(A)sin(B) + cos(A)cos(B)] * sin(B)

sinAcos^2B - cosAsinBcosB + sinAsin^2B + cosAcosBsinB

sinAcos^2B + sinAsin^2B

sinA(sin^2B + cos^2B)

sinA(1)

sinA

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- PatrickLv 68 years ago
Recall:

sin(x - y) = sin(x)cos(y) - cos(x)sin(y);

cos(x - y) = cos(x)cos(y) + sin(x)sin(y);

cos(x)^2 + sin(x)^2 = 1.

sin(a - b)cos(b) + cos(a - b)sin(b)

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

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

= sin(a)cos(b)^2 + sin(a)sin(b)^2

= sin(a)(cos(b)^2 + sin(b)^2)

= sin(a).

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