16. $$ \sin x \cos x \cos 2 x \cos 4 x $$ 17. $$ \sin (A+B) \cdot \sin (A-B) $$

Question

UpStudy AI · Accepted Answer

To simplify the given expressions, we can use trigonometric identities.

1. $$ \sin x \cos x \cos 2 x \cos 4 x $$

Using the double angle formula for cosine, we have:
$$ \cos 2x = 2\cos^2 x - 1 $$
$$ \cos 4x = 2\cos^2 2x - 1 $$

Substituting these into the expression, we get:
$$ \sin x \cos x (2\cos^2 x - 1)(2(2\cos^2 x - 1)^2 - 1) $$

2. $$ \sin (A+B) \cdot \sin (A-B) $$

Using the sum and difference formulas for sine, we have:
$$ \sin (A+B) = \sin A \cos B + \cos A \sin B $$
$$ \sin (A-B) = \sin A \cos B - \cos A \sin B $$

Substituting these into the expression, we get:
$$ (\sin A \cos B + \cos A \sin B)(\sin A \cos B - \cos A \sin B) $$

Now, we can simplify these expressions further if needed.
1. $$ \sin x \cos x \cos 2x \cos 4x $$ simplifies to $$ \frac{\sin 8x}{16} $$. 2. $$ \sin (A+B) \cdot \sin (A-B) $$ simplifies to $$ \frac{\cos 2B - \cos 2A}{2} $$.

16. \( \sin x \cos x \cos 2 x \cos 4 x \) 17. \( \sin (A+B) \cdot \sin (A-B) \)

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Trigonometry Questions