Question

COS (X-y) = cos(x-Y) = 2 cos x cosy​

Answer 1

Answer:

Remember your formula:

cos

(

x

+

y

)

=

(

cos

x

⋅

cos

y

)

−

(

sin

x

⋅

sin

y

)

Now, try this:

cos

(

x

−

y

)

=

cos

(

x

+

(

−

y

)

)

...so you can apply your formula again:

=

cos

x

⋅

cos

(

−

y

)

−

sin

x

⋅

sin

(

−

y

)

Now here's the trick: remember that cosine is a symmetrical function about x = 0. This means that cos(-y) = cos(y) for all y.

Sine, however, is NOT symmetrical. sin(-y) = -sin(y) for all y.

(look at the graphs of these functions to verify this).

So you can rewrite

cos

(

x

−

y

)

as:

cos

x

⋅

cos

y

−

(

sin

x

⋅

(

−

sin

y

)

)

=

(

cos

x

⋅

cos

y

)

+

(

sin

x

⋅

sin

y

)

So therefore:

cos

(

x

+

y

)

+

cos

(

x

−

y

)

=

(

(

cos

x

⋅

cos

y

)

−

(

sin

x

⋅

sin

y

)

)

+

(

(

cos

x

⋅

cos

y

)

+

(

sin

x

⋅

sin

y

)

)

=

(

cos

x

⋅

cos

y

)

+

(

cos

x

⋅

cos

y

)

=

2

(

cos

x

⋅

cos

y

)