Inverse Trigonometric Identities

Step 1: Use a trigonometric identity.
We will use the identity for the sum of arctangents: $%5Ctan%5E%7B-1%7D%20a%20%2B%20%5Ctan%5E%7B-1%7D%20b%20%3D%20%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7Ba%20%2B%20b%7D%7B1%20-%20ab%7D%20%5Cright)$ Here, we are given $2%20%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B1%7D%7B3%7D%20%5Cright)$ , so let $%5Ctheta%20%3D%20%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B1%7D%7B3%7D%20%5Cright)$ , meaning $%5Ctan%20%5Ctheta%20%3D%20%5Cfrac%7B1%7D%7B3%7D$ . Then, using the double angle formula for tangent: $2%5Ctheta%20%3D%20%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B2%20%5Ccdot%20%5Cfrac%7B1%7D%7B3%7D%7D%7B1%20-%20%5Cleft(%20%5Cfrac%7B1%7D%7B3%7D%20%5Cright)%5E2%7D%20%5Cright)$ $2%5Ctheta%20%3D%20%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B%5Cfrac%7B2%7D%7B3%7D%7D%7B%5Cfrac%7B8%7D%7B9%7D%7D%20%5Cright)%20%3D%20%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B3%7D%7B4%7D%20%5Cright)$

Step 2: Add the inverse cosine term.
Now, we are asked to prove: $2%20%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B1%7D%7B3%7D%20%5Cright)%20%2B%20%5Ccos%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B3%7D%7B5%7D%20%5Cright)$ Using the relationship between tangent and cosine, and recognizing that $%5Ccos%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B3%7D%7B5%7D%20%5Cright)$ corresponds to the angle whose cosine is $%5Cfrac%7B3%7D%7B5%7D$ , we conclude: $2%20%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B1%7D%7B3%7D%20%5Cright)%20%2B%20%5Ccos%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B3%7D%7B5%7D%20%5Cright)%20%3D%20%5Cfrac%7B%5Cpi%7D%7B2%7D$

Final Answer: $%5Cboxed%7B2%20%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B1%7D%7B3%7D%20%5Cright)%20%2B%20%5Ccos%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B3%7D%7B5%7D%20%5Cright)%20%3D%20%5Cfrac%7B%5Cpi%7D%7B2%7D%7D$

High-Yield Trend

Chapter Questions
1 MCQs

Official Solution

Inverse Trigonometric Identities

High-Yield Trend

Chapter Questions 1 MCQs

Official Solution

Chapter Questions
1 MCQs