Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of the expression. Do not use a calculator.
tangent 50 degrees minus cotangent 40 degrees
Question content area bottom
Part 1
tangent 50 degrees minus cotangent 40 degreesequals
enter your response here
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
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.