Question Video: Using Pythagorean Identities to Evaluate Trigonometric Expressions Mathematics

Video Transcript

Find the value of cot 𝜃 minus csc 𝜃 divided by tan 𝜃 minus sec 𝜃 given 𝜃 exists in the open interval from three 𝜋 over two to two 𝜋 and sin 𝜃 is equal to negative four-fifths.

Using our knowledge of the CAST diagram, since 𝜃 lies between three 𝜋 over two and two 𝜋 radians, we know it lies in the fourth quadrant. We know that the cosine of any angle in this quadrant is positive, whereas the sine and tangent of any angle here are negative. This ties in with the fact that we are told that sin 𝜃 is equal to negative four-fifths. We can use this information to sketch a right triangle in the fourth quadrant as shown. This triangle is a Pythagorean triple consisting of three positive integers three, four, and five such that three squared plus four squared is equal to five squared. We know that cos 𝛼 is equal to the adjacent over the hypotenuse, and tan 𝛼 is equal to the opposite over the adjacent.

From our diagram, we therefore have cos 𝛼 is equal to three-fifths and tan 𝛼 is equal to four-thirds. The angle 𝜃 is equal to 360 degrees minus 𝛼. And using the properties of related angles, we know that the cos of 360 degrees minus 𝛼 is equal to cos 𝛼. This means that cos 𝜃 is equal to cos 𝛼, which is equal to three-fifths. Likewise, we know that the tan of 360 degrees minus 𝛼 is equal to negative tan 𝛼. tan 𝜃 is therefore equal to negative tan 𝛼, which is equal to negative four-thirds. This ties in with the fact that the cosine of any angle is positive and the tangent of any angle is negative in the fourth quadrant.

Our next step is to recall the reciprocal trigonometric identities. csc 𝜃 is equal to one over sin 𝜃, sec 𝜃 is equal to one over cos 𝜃, and cot 𝜃 is equal to one over tan 𝜃. The three functions are the reciprocals of sin 𝜃, cos 𝜃, and tan 𝜃, respectively. This means that csc 𝜃 is equal to negative five-quarters, sec 𝜃 is equal to five-thirds, and cot 𝜃 is equal to negative three-quarters.

We are now in a position to substitute our values into the expression cot 𝜃 minus csc 𝜃 divided by tan 𝜃 minus sec 𝜃. On the numerator, we have negative three-quarters minus negative five-quarters. And on the denominator, we have negative four-thirds minus five-thirds. The numerator simplifies to two-quarters or one-half, and the denominator becomes negative nine over three, which is equal to negative three. Dividing one-half by negative three gives us negative one-sixth. If 𝜃 lies between three 𝜋 over two and two 𝜋 and sin 𝜃 is equal to negative four-fifths, then cot 𝜃 minus csc 𝜃 divided by tan 𝜃 minus sec 𝜃 is equal to negative one-sixth.