Video: Evaluating Trigonometric Expressions Using Sum of Angles Identities Involving Special Angles

Simplify (tan(118Β° βˆ’ 2𝑋) + tan(32Β° + 2𝑋))/(1 βˆ’ tan(118Β° βˆ’ 2𝑋) tan(32Β° + 2𝑋)).

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Video Transcript

Simplify tan of 118 degrees minus two 𝑋 plus tan of 32 degrees plus two 𝑋 all over one minus tan of 118 degrees minus two 𝑋 times tan of 32 degrees plus two 𝑋.

To simplify this, we’ll need to recognize a pattern. To help us recognize this pattern, we’re going to take what’s in the parentheses and represent that as a variable. We can say that 𝑋 equals 118 degrees minus two 𝑋 and that 𝑦 equals 32 degrees plus two 𝑋.

We’ll then have something like this, the tan of 𝑋 plus the tan of 𝑦 over one minus tan 𝑋 times tan 𝑦. And this is a form of a trig identity. It’s a form of tan of 𝛼 plus 𝛽. Tan of 𝛼 plus 𝛽 equals tan of 𝛼 plus tan of 𝛽 over one minus tan of 𝛼 times tan of 𝛽. If we apply that to our expression, we can simplify it to say tan of 𝑋 plus 𝑦.

We’ll take our values for 𝑋 and 𝑦 and plug them back in. And now we have tan of 118 degrees minus two 𝑋 plus 32 degrees plus two 𝑋. Minus two 𝑋 plus two 𝑋 cancels out. 118 degrees plus 32 degrees equals 150 degrees. Tan of 150 degrees is equal to negative square root of three over three.

The most simplified form of this expression is negative square root of three over three.

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