How do you simplify cos(45^@+a)cos(45^@-a)-sin(45^@-a)sin(45^@+a) ?

1 Answer
May 10, 2018

0

Explanation:

Yes, it simplifies to give you zero. Now to make our lives easier, it would be better to turn the equation to:

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

This way we can resolve both sides without things getting mixed and muddled together.

You should be comfortable with your trigonometric identities so by looking at this equation you are able to tell the four identities we need to know on both LHS and RHS.

sin(a+b) = sinacosb + sinbcosa
sin(a-b) = sinacosb - sinbcosa
cos(a+b) = cosacosb - sinasinb
cos(a-b) = cosacosb + sinasinb

LHS (lefthandside):

cos(45+a) = cos45cosa - sin45sina
cos(45-a) = cos45cosa + sin45sina

cos45 put in the calculator will give you 1/sqrt2 which is the same value given for sin45

cos(45+a) = 1/sqrt2cosa - 1/sqrt2sina
cos(45+a) = 1/sqrt2(cosa - sina)

cos(45-a) = 1/sqrt2cosa + 1/sqrt2sina
cos(45-a) = 1/sqrt2(cosa + sina)

cos(45+a)cos(45-a) = 1/sqrt2(cosa + sina) times 1/sqrt2(cosa - sina)

= 1/sqrt2(cosa + sina) times 1/sqrt2(cosa - sina)

= 1/2(cosa+sina)(cosa-sina)

= 1/2(cos^2a-sin^2a)

Now that's the Left Hand Side Done.

RHS (righthandside):
follow the same steps as the LHS
sin(45+a) = sin45cosa + sinacos45
sin(45+a) = 1/sqrt2(cosa+sina)

sin(45-a) = sin45cosa - sinacos45
sin(45-a) = 1/sqrt2(cosa-sina)

sin(45-a)sin(45+a) = 1/sqrt2(cosa-sina) times 1/sqrt2(cosa+sina)

= 1/sqrt2(cosa-sina) times 1/sqrt2(cosa+sina)

= 1/2(cosa-sina)(cosa+sina)

= 1/2(cos^2a - sin^2a)

This is the same simplified value given for the LHS

If the LHS gives the same value as the RHS, this will simplify to give you zero.