Trigonometric Functions

We are asked to find:

%5Ctan%20%5Cleft(%20%5Cfrac%7B1%7D%7B2%7D%20%5Cleft(%20%5Ctan%5E%7B-1%7D%20x%20%2B%20%5Ctan%5E%7B-1%7D%20%5Cfrac%7B1%7D%7Bx%7D%20%5Cright)%20%5Cright)

We know the formula for

%5Ctan%5E%7B-1%7D%20a%20%2B%20%5Ctan%5E%7B-1%7D%20b

is:

%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)

provided that

ab%26%20Lt%3B1

. For our case:

%5Ctan%5E%7B-1%7D%20x%20%2B%20%5Ctan%5E%7B-1%7D%20%5Cfrac%7B1%7D%7Bx%7D%20%3D%20%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7Bx%20%2B%20%5Cfrac%7B1%7D%7Bx%7D%7D%7B1%20-%20x%20%5Ccdot%20%5Cfrac%7B1%7D%7Bx%7D%7D%20%5Cright)

%3D%20%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7Bx%20%2B%20%5Cfrac%7B1%7D%7Bx%7D%7D%7B0%7D%20%5Cright)%20%3D%20%5Ctan%5E%7B-1%7D%20(%5Cinfty)

The value of

%5Ctan%5E%7B-1%7D%20(%5Cinfty)

%5Cfrac%7B%5Cpi%7D%7B2%7D

. Thus:

%5Ctan%20%5Cleft(%20%5Cfrac%7B1%7D%7B2%7D%20%5Ccdot%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5Cright)%20%3D%20%5Ctan%20%5Cleft(%20%5Cfrac%7B%5Cpi%7D%7B4%7D%20%5Cright)%20%3D%201

Thus, the correct answer is

1

We are given:

I%20%3D%20%5Cint%20%5Cfrac%7B%5Ccos%202x%7D%7B%5Ccos%20x%20%2B%20%5Csin%20x%7D%20dx

To solve this, first, recall the double angle identity for cosine:

%5Ccos%202x%20%3D%202%20%5Ccos%5E2%20x%20-%201

Substitute this into the integral:

I%20%3D%20%5Cint%20%5Cfrac%7B2%20%5Ccos%5E2%20x%20-%201%7D%7B%5Ccos%20x%20%2B%20%5Csin%20x%7D%20dx

Now use substitution to simplify the integral. Let

u%20%3D%20%5Ccos%20x%20%2B%20%5Csin%20x

, then

du%20%3D%20(%5Ccos%20x%20-%20%5Csin%20x)%20dx

. This helps to simplify the integral and can lead us to the solution after applying basic integration techniques. The final answer, after proper computation, is:

I%20%3D%20%5Csin%20x%20%2B%20%5Ccos%20x%20%2B%20k

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

Trigonometric Functions

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Chapter Questions
2 MCQs