Trig Expression simplification

[cbarker1]

What identity I need to use for simplifying this trig expression into one expression?$cos(3x)+4hcos(x)$ where h is a constant.

Thank you for your help.

Can you explain, too?

 

[anemone]

To simplify an expression that contains two terms means we need to combine these two terms to become one single term, i.e. by factoring out the common factor.

In our case ($\cos3x+4h \cos x$), we need to express $\cos 3x$ in terms of $\cos x$ since the second term has a $\cos x$ in it.

By using the triple-angle formula for $ \cos 3x$, where $ \cos 3x=4\cos^3x-3 \cos x$, we can simplify the original expression as follows:

$\displaystyle \cos3x+4h \cos x =(4\cos^3x-3 \cos x)+4h \cos x=\cos x(4\cos^2x-3+4h)$

 

[cbarker1]

I wished to use sum-product identity.

 

[MarkFL]

With differing coefficients on the cosine terms, I don't see how you can use a sum-to-product identity.

 

[CellCoree]

To simplify this trig expression into one expression, you can use the trigonometric identity: cos(a+b) = cos(a)cos(b) - sin(a)sin(b).

This identity allows you to rewrite cos(3x) as cos(x+x+x), which can then be expanded using the identity above. Similarly, you can use the identity cos(2x) = cos^2(x) - sin^2(x) to simplify cos(x)cos(x) in the expression.

Overall, the simplified expression would be cos(x)(cos^2(x) - sin^2(x)) + 4hcos(x). This can be further simplified using the Pythagorean identity cos^2(x) + sin^2(x) = 1, resulting in the final expression of cos(x) + 4hcos(x) = cos(x)(1+4h).

I hope this explanation helps. Please let me know if you have any further questions.