Use Fundamental Identities​ and/or the Complementary Angle Theorem to find the exact value of the expression. Do not use a calculator.

Question

tangent 50 degrees minus cotangent 40 degrees
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Part 1
tangent 50 degrees minus cotangent 40 degreesequals
  
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​(Simplify your​ answer, including any radicals. Use integers or fractions for any numbers in the​ expression.)

Answer 1

First, we can use the Fundamental Identity for Tangent to rewrite tangent 50 degrees:

tangent 50 degrees = sin 50 degrees / cos 50 degrees

And we can use the Fundamental Identity for Cotangent to rewrite cotangent 40 degrees:

cotangent 40 degrees = cos 40 degrees / sin 40 degrees

Now, we can substitute these into the expression:

sin 50 degrees / cos 50 degrees - cos 40 degrees / sin 40 degrees

Next, we can find a common denominator:

(sin 50 degrees * sin 40 degrees) / (cos 50 degrees * sin 40 degrees) - (cos 40 degrees * cos 50 degrees) / (sin 40 degrees * cos 50 degrees)

This simplifies to:

(sin 50 degrees * sin 40 degrees - cos 40 degrees * cos 50 degrees) / (cos 50 degrees * sin 40 degrees)

Finally, we can use the Complementary Angle Theorem to simplify this expression further:

sin 50 degrees * sin 40 degrees = cos(90 - 50) * sin(90 - 40) = cos 40 degrees * cos 50 degrees

Therefore, the expression simplifies to:

(cos 40 degrees * cos 50 degrees - cos 40 degrees * cos 50 degrees) / (cos 50 degrees * sin 40 degrees) = 0 / (cos 50 degrees * sin 40 degrees) = 0

So, tangent 50 degrees minus cotangent 40 degrees equals 0.