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Write the expression as the sine or cosine of an angle.

[tex]\[
\cos 7y \cos 3y - \sin 7y \sin 3y
\][/tex]

Hint:

[tex]\[
\begin{aligned}
\sin (A \pm B) &= \sin A \cos B \pm \cos A \sin B \\
\cos (A \pm B) &= \cos A \cos B \mp \sin A \sin B
\end{aligned}
\][/tex]

[tex]\[
\cos (7y - 3y)
\][/tex]

Sagot :

GinnyAnsweridn
Sure, let's carefully apply the relevant trigonometric identities to simplify the given expression:

Given expression:
[tex]\[ \cos(7y) \cos(3y) - \sin(7y) \sin(3y) \][/tex]

We can use the cosine angle addition formula:
[tex]\[ \cos(A + B) = \cos(A) \cos(B) - \sin(A) \sin(B) \][/tex]

By comparing this formula with our given expression [tex]\(\cos(7y) \cos(3y) - \sin(7y) \sin(3y)\)[/tex], we see that it fits perfectly if we let [tex]\( A = 7y \)[/tex] and [tex]\( B = 3y \)[/tex]. So,

[tex]\[ \cos(7y) \cos(3y) - \sin(7y) \sin(3y) = \cos((7y) + (3y)) \][/tex]

Therefore, the given expression simplifies to:
[tex]\[ \cos(10y) \][/tex]

So, the simplified form of [tex]\(\cos(7y) \cos(3y) - \sin(7y) \sin(3y)\)[/tex] is:
[tex]\[ \cos(10y) \][/tex]
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