Using Power Reducing Formulas to rewrite Trig Expressions

In summary: The answer given is:##2-\cos^2 (2x)\ ## has a cosine function with a power of 2 .Looking into your reply Mark! Give me a few moments to track down my mistake.And thanks for that Sammy. So many twos floating around they all seem to blend into each other.Some progress I think. I've got it down to the power of 1, but can't seem to put my finger on this. Feels like untangling a spool of fishing line.##2-cos^22x####=2-\frac{1+cos4x}{2}####=\frac{4}{2}-\frac{1
  • #1
opus
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Homework Statement


Use the power reducing formulas to rewrite the expression that does not contain trigonometric functions of power greater than 1.

Given expression:
##4sin^2xcos^2x##

2. Homework Equations

Relevant Power-Reducing Formulas:
##sin^2x=\frac{1-cos2x}{2}##
##cos^2x=\frac{1+cos2x}{2}##

The Attempt at a Solution



$$4sin^2xcos^2x$$
$$=4\left(\frac{1-cos2x}{2}\right)\left(\frac{1+cos2x}{2}\right)$$
$$=4\left(\frac{1+cos2x-cos2x-cos^22x}{2}\right)$$
$$=4\left(\frac{1-cos^22x}{2}\right)$$
$$=2-cos^22x$$

The answer given is:
##\frac{1-cos4x}{2}##

Have I solved done a miscalculation somewhere, or is my entire approach to solving this wrong?
Thank you for any responses.
 
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  • #2
opus said:

Homework Statement


Use the power reducing formulas to rewrite the expression that does not contain trigonometric functions of power greater than 1.

Given expression:
##4sin^2xcos^2x##

2. Homework Equations

Relevant Power-Reducing Formulas:
##sin^2x=\frac{1-cos2x}{2}##
##cos^2x=\frac{1+cos2x}{2}##

The Attempt at a Solution



$$4sin^2xcos^2x$$
$$=4\left(\frac{1-cos2x}{2}\right)\left(\frac{1+cos2x}{2}\right)$$
$$=4\left(\frac{1+cos2x-cos2x-cos^22x}{2}\right)$$
$$=4\left(\frac{1-cos^22x}{2}\right)$$
$$=2-cos^22x$$
You have a mistake in the line above. Fix the mistake, and then use one of your formulas to reduce the power of the ##\cos^2(2x)## term.
opus said:
The answer given is:
##\frac{1-cos4x}{2}##

Have I solved done a miscalculation somewhere, or is my entire approach to solving this wrong?
Thank you for any responses.
 
  • #3
opus said:

Homework Statement


Use the power reducing formulas to rewrite the expression that does not contain trigonometric functions of power greater than 1.

Given expression:
##4sin^2xcos^2x##

2. Homework Equations

Relevant Power-Reducing Formulas:
##sin^2x=\frac{1-cos2x}{2}##
##cos^2x=\frac{1+cos2x}{2}##

The Attempt at a Solution


$$4sin^2xcos^2x$$ $$=4\left(\frac{1-cos2x}{2}\right)\left(\frac{1+cos2x}{2}\right)$$ $$=4\left(\frac{1+cos2x-cos2x-cos^22x}{2}\right)$$ $$=4\left(\frac{1-cos^22x}{2}\right)$$ $$=2-cos^22x$$
The answer given is:
##\frac{1-cos4x}{2}##

Have I solved done a miscalculation somewhere, or is my entire approach to solving this wrong?
Thank you for any responses.
##2-\cos^2 (2x)\ ## has a cosine function with a power of 2 .
 
  • #4
Looking into your reply Mark! Give me a few moments to track down my mistake.
And thanks for that Sammy. So many twos floating around they all seem to blend into each other.
 
  • #5
Some progress I think. I've got it down to the power of 1, but can't seem to put my finger on this. Feels like untangling a spool of fishing line.
##2-cos^22x##
##=2-\frac{1+cos4x}{2}##
##=\frac{4}{2}-\frac{1+cos4x}{2}##=##\frac{3+cos4x}{2}##

Mark which like were you talking about where I made the error? Was it the last line in my post? The ##2-cos^22x##?
 
  • #6
opus said:
Some progress I think. I've got it down to the power of 1, but can't seem to put my finger on this. Feels like untangling a spool of fishing line.
##2-cos^22x##
##=2-\frac{1+cos4x}{2}##
##=\frac{4}{2}-\frac{1+cos4x}{2}##=##\frac{3+cos4x}{2}##

Mark which like were you talking about where I made the error? Was it the last line in my post? The ##2-cos^22x##?
Yes, this one.
The preceding line has ##4(\frac{1-cos^22x}{2})##. This is equal to ##2(1 - \cos^2(2x)) \ne 2 - \cos^2(2x)##
 
  • #7
opus said:
$$=4\left(\frac{1-cos2x}{2}\right)\left(\frac{1+cos2x}{2}\right)$$
$$=4\left(\frac{1+cos2x-cos2x-cos^22x}{2}\right)$$
You also have an error between these two lines.
 
  • #8
There is also a well known formula about ##sin~x.cos~x## which it would have been my first step to use.
 

Related to Using Power Reducing Formulas to rewrite Trig Expressions

1. How do I know when to use a power reducing formula to rewrite a trig expression?

Power reducing formulas are typically used when the trig expression has a higher power, such as sin^2(x) or cos^2(x). These formulas can help simplify the expression and make it easier to solve.

2. Can I use a power reducing formula for any trig expression?

No, power reducing formulas are specifically designed for expressions with squared trig functions, such as sin^2(x) or cos^2(x). They cannot be used for other types of trig expressions.

3. What is the most commonly used power reducing formula for rewriting trig expressions?

The most commonly used power reducing formula is sin^2(x) + cos^2(x) = 1. This formula is also known as the Pythagorean identity and is used to simplify many trig expressions.

4. How do I know if I have correctly rewritten a trig expression using a power reducing formula?

You can check your answer by substituting the original expression with your rewritten expression and simplifying. If the simplified expressions are equal, then you have correctly used the power reducing formula.

5. Can I use more than one power reducing formula to rewrite a trig expression?

Yes, you can use multiple power reducing formulas in succession to simplify a trig expression. Just be sure to keep track of the changes you make and check your final answer for accuracy.

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