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Find the solution set of tan 𝑥 plus tan seven plus tan 𝑥 tan seven equals one, where 𝑥 lies between zero and 360 degrees inclusive.
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In order to solve this problem, we’ll use the property that tan of 𝐴 plus 𝐵 is equal to tan 𝐴 plus tan 𝐵 divided by one minus tan 𝐴 multiplied by tan 𝐵.
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Our equation can be rewritten as tan 𝑥 plus tan seven is equal to one minus tan 𝑥 multiplied by tan seven.
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Dividing both sides by one minus tan 𝑥 multiplied by tan seven gives us tan 𝑥 plus tan seven divided by one minus tan 𝑥 multiplied by tan seven is equal to one.
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This is now written in the same form as the general equation, where 𝐴 is equal to 𝑥 and 𝐵 is equal to seven.
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This means that tan of 𝑥 plus seven must be equal to one.
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The angle whose tangent is equal to one in the first quadrant is 45 degrees and the angle whose tangent is equal to one in the third quadrant is equal to 225 degrees.
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This means that tan of 𝑥 plus seven is equal to tan 45 and also tan of 𝑥 plus seven is equal to tan of 225.
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Solving the first equation gives us 𝑥 plus seven is equal to 45.
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Therefore, 𝑥 is equal to 38 degrees.
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The second equation gives us 𝑥 plus seven is equal to 225.
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Therefore, 𝑥 is equal to 218 degrees.
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The solution set of tan 𝑥 plus tan seven plus tan 𝑥 multiplied by tan seven equals one is 𝑥 equals 38 degrees and 𝑥 equals 218 degrees.