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Exercises Trigonometry

Example 3 from Algebra and Trigonometry, 9.3 Double-Angle, Half-Angle, and Reduction Formulas

Using the Double-Angle Formulas to Verify an Identity

Verify the following identity using double-angle formulas:

1 + \sin (2 θ) = {(\sin θ + \cos θ)}^{2}

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We will work on the right side of the equal sign and rewrite the expression until it matches the left side.

$\begin{matrix} {(\sin θ + \cos θ)}^{2} & = & \sin^{2} θ + 2 \sin θ \cos θ + \cos^{2} θ \\ = & (\sin^{2} θ + \cos^{2} θ) + 2 \sin θ \cos θ \\ = & 1 + 2 \sin θ \cos θ \\ = & 1 + \sin (2 θ) \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Verify the identity: $\cos^{4} θ - \sin^{4} θ = \cos (2 θ) .$

How did it go?

PreviousUsing the Double-Angle Formula for Cosine without Exact Values Next Verifying a Double-Angle Identity for Tangent

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 9.3, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.