Trigonometric Identities

Step 1: {Evaluate each term}

%5Ctan%5E%7B-1%7D(1)%20%3D%20%5Cfrac%7B%5Cpi%7D%7B4%7D%2C%20%5Cquad%20%5Ccos%5E%7B-1%7D%5Cleft(-%5Cfrac%7B1%7D%7B2%7D%5Cright)%20%3D%20%5Cpi%20-%20%5Cfrac%7B%5Cpi%7D%7B3%7D%2C%20%5Cquad%20%5Csin%5E%7B-1%7D%5Cleft(-%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%5Cright)%20%3D%20-%5Cfrac%7B%5Cpi%7D%7B4%7D.

Step 2: {Add the terms}

%5Cfrac%7B%5Cpi%7D%7B4%7D%20%2B%20%5Cleft(%5Cpi%20-%20%5Cfrac%7B%5Cpi%7D%7B3%7D%5Cright)%20-%20%5Cfrac%7B%5Cpi%7D%7B4%7D.

Simplify:

%5Cfrac%7B%5Cpi%7D%7B4%7D%20%2B%20%5Cpi%20-%20%5Cfrac%7B%5Cpi%7D%7B3%7D%20-%20%5Cfrac%7B%5Cpi%7D%7B4%7D%20%3D%20%5Cfrac%7B2%5Cpi%7D%7B3%7D.

Conclusion: The principal value is

%5Cfrac%7B2%5Cpi%7D%7B3%7D

Step 1: {Simplify the expression inside

%5Ctan%5E%7B-1%7D

}
The given expression is:

%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B%5Ccos%20x%7D%7B1%20-%20%5Csin%20x%7D%20%5Cright).

Using trigonometric identities, rewrite:

1%20-%20%5Csin%20x%20%3D%20(%5Ccos%5E2%5Cfrac%7Bx%7D%7B2%7D%20%2B%20%5Csin%5E2%5Cfrac%7Bx%7D%7B2%7D)%20-%202%5Csin%5Cfrac%7Bx%7D%7B2%7D%5Ccos%5Cfrac%7Bx%7D%7B2%7D%20%3D%20(%5Ccos%5Cfrac%7Bx%7D%7B2%7D%20-%20%5Csin%5Cfrac%7Bx%7D%7B2%7D)%5E2.

Step 2: {Transform into a single tangent function}
Substituting

1%20-%20%5Csin%20x

and

%5Ccos%20x%20%3D%20(%5Ccos%5E2%5Cfrac%7Bx%7D%7B2%7D%20-%20%5Csin%5E2%5Cfrac%7Bx%7D%7B2%7D)

%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B%5Ccos%20x%7D%7B1%20-%20%5Csin%20x%7D%20%5Cright)%0A%3D%20%5Ctan%5E%7B-1%7D%20%5Cleft%5B%20%5Ctan%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D%20%2B%20%5Cfrac%7Bx%7D%7B2%7D%5Cright)%20%5Cright%5D.

Step 3: {Simplify using

%5Ctan%5E%7B-1%7D%20%5Ctan%20y%20%3D%20y

}
Since

-%5Cfrac%7B%5Cpi%7D%7B2%7D%3Cx%3C%5Cfrac%7B%5Cpi%7D%7B2%7D

, we simplify:

%5Ctan%5E%7B-1%7D%20%5Cleft%5B%20%5Ctan%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D%20%2B%20%5Cfrac%7Bx%7D%7B2%7D%5Cright)%20%5Cright%5D%20%3D%20%5Cfrac%7B%5Cpi%7D%7B4%7D%20%2B%20%5Cfrac%7Bx%7D%7B2%7D.

Conclusion: The simplest form is

%5Cfrac%7B%5Cpi%7D%7B4%7D%20%2B%20%5Cfrac%7Bx%7D%7B2%7D

Chapter Questions
2 MCQs