Express cos9x+cos5x as a product.
Can you just multiply the 9 and 5 to get: cos45x
? is that a product?
oohhhh noooo!
we have a formula
cosA + cosB = 2cos(A+B)/2 * cos(A-B)/2
so cos 9x + cos 5x = 2cos(9x+5x)/2 cos(9x-5x)/2
= 2(cos 7x)(cos 2x)
No, simply multiplying the exponents of the trigonometric functions does not result in a valid product. However, there is a trigonometric identity called the double angle formula that can be applied here. The double angle formula for cosine states:
cos(2θ) = 2cos²(θ) - 1
Using this formula, we can write cos9x as:
cos9x = cos(2 * 4.5x)
= 2cos²(4.5x) - 1
Similarly, cos5x can be written as:
cos5x = cos(2 * 2.5x)
= 2cos²(2.5x) - 1
Now we can rewrite the expression cos9x + cos5x by substituting the above expressions:
cos9x + cos5x = (2cos²(4.5x) - 1) + (2cos²(2.5x) - 1)
Next, we can expand the equation and simplify:
cos9x + cos5x = 2cos²(4.5x) + 2cos²(2.5x) - 2
Therefore, the expression cos9x + cos5x can be expressed as a product of the cosine functions squared minus 2.