Video: Using Angle-Sum and -Difference Identities to Simplify Trigonometric Expressions

Simplify cos 2𝑋 cos 22𝑋 − sin 2𝑋 sin 22𝑋.

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Video Transcript

Simplify cos of two 𝑋 times cos of 22𝑋 minus sin of two 𝑋 times sin of 22𝑋.

When we look at this expression, we see that we have the angle two 𝑋 and the angle 22𝑋. We have the cos and sin of the angle two 𝑋, and we have the cos and sin of the angle 22𝑋, which means we have an expression in the form cos of 𝐴 times cos of 𝐵 minus sin of 𝐴 times sin of 𝐵. And we can simplify this expression using the angle sum identities. We know that the cos of 𝐴 times the cos of 𝐵 minus the sin of 𝐴 times the sin of 𝐵 is equal to the cos of 𝐴 plus 𝐵. In our case, the angle 𝐴 is two 𝑋 and the angle 𝐵 is 22𝑋, making this expression equal to cos of two 𝑋 plus 22𝑋, cos of 24𝑋. And so the simplified form of this expression is cos of 24𝑋.

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