Video: Using the Sum and Difference of Angles Identities to Evaluate a Trigonometric Expression Involving Special Angles

Evaluate (tan (5𝜋/6) − tan (2𝜋/3))/(1 + tan (5𝜋/6) tan (2𝜋/3)).

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Video Transcript

Evaluate tan of five 𝜋 over six minus tan of two 𝜋 over three all divided by one plus tan five 𝜋 over six multiplied by tan two 𝜋 over three.

Before starting this question, we recall one of our compound angle identities. Tan 𝐴 minus 𝐵 is equal to tan 𝐴 minus tan 𝐵 divided by one plus tan 𝐴 multiplied by tan 𝐵. In this question, we see that angle 𝐴 is equal to five 𝜋 over six. Angle 𝐵 is equal to two 𝜋 over three. This means that our expression is equal to tan of five 𝜋 over six minus two 𝜋 over three. Multiplying the numerator and denominator of the second angle by two gives us four 𝜋 over six.

Five 𝜋 over six minus four 𝜋 over six is equal to 𝜋 over six. 𝜋 over six radians is equal to 30 degrees. And this is one of our standard trig angles. Tan of 30 degrees is equal to root three over three. So tan of 𝜋 over six radians is also equal to root three over three. We can, therefore, conclude that tan five 𝜋 over six minus tan two 𝜋 over three divided by one plus tan five 𝜋 over six multiplied by tan two 𝜋 over three is equal to root three over three.

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