Question

How do I move one side to the other: n%5E2%5Calpha%20-%201" id="TexFormula1" title="sin^2\alpha - cos^2\alpha = 2sin^2\alpha - 1" alt="sin^2\alpha - cos^2\alpha = 2sin^2\alpha - 1" align="absmiddle" class="latex-formula">

Answer 1

9514 1404 393

Explanation:

You can use the addition property of equality to "move" a term from one side of the equation to the other. For example, if you want to move the cos²α term, you can add cos²α to both sides of the equation:

  sin²α -cos²α +cos²α = 2sin²α -1 +cos²α

When this is simplified, it becomes ...

  sin²α = 2sin²α +cos²α -1 . . . . . . the cos²α term is gone from the left

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The equation you have is an identity. The left and right sides are equal for any value of α. When you have such an equation as a trig problem, you are often being asked to prove it is true. The way you do that is to make use of other trig identities to transform one side so that it matches the other side.

Here, you can use the trig identity ...

  sin²α +cos²α = 1

If you use this to substitute for 1 on the right, you have ...

  sin²α - cos²α = 2sin²α - (sin²α +cos²α)

Now, when you collect terms, you get ...

  sin²α - cos²α = 2sin²α - sin²α -cos²α . . . eliminate parentheses

  sin²α - cos²α = sin²α - cos²α . . . . . proof of your identity