Step 1: $=\frac{\sin (x-y)}{\cos (x-y)}$
Step 2: $\frac{\sin (x) \cos (y)+\cos (x) \sin (y)}{(A) \cos (y)-\sin (x) \sin (y)}$
Step 3: $\frac{\frac{\sin (x) \cos (y)+\cos (x) \sin (y)}{\cos (x)(B)}}{\frac{\cos (x) \cos (y)-\sin (x) \sin (y)}{\cos (x) \cos (y)}}$
Step 4: $\frac{\frac{\sin (x) \cos (y)}{\cos (x) \cos (y)}+\frac{\cos (x) \sin (y)}{\cos (x) \cos (y)}}{\frac{\cos (x) \cos (y)}{\cos (x) \cos (y)}-\frac{\sin (x) \sin (y)}{\cos (x) \cos (y)}}$
Step 5: $\frac{\tan (x)+\tan y}{1-\tan (x)(c)}$
The work shown is a way to derive
$\tan \left(x+y=\frac{\tan (x)+\tan (y)}{1-\tan (x) \tan (y)}\right.$
What expressions go in the derivation of the tangent sum identity in place of $A, B$, and $C$?
A:
B:
C:
Answer (2)
A is cos ( x ) .
B is cos ( y ) .
C is tan ( y ) .
The expressions that go in the derivation of the tangent sum identity in place of A, B, and C are A = cos ( x ) , B = cos ( y ) , C = tan ( y ) .
Explanation
Problem Analysis We are given a derivation of the tangent sum identity and need to find the expressions that fit in the places of A, B, and C. Let's analyze the steps.
Step 1 Analysis Step 1 presents the tangent function as a ratio of sine and cosine: tan ( x + y ) = cos ( x + y ) sin ( x + y )
Step 2 Analysis: Finding A Step 2 expands the sine and cosine of the sum of angles: cos ( x + y ) sin ( x + y ) = cos ( x ) cos ( y ) − sin ( x ) sin ( y ) sin ( x ) cos ( y ) + cos ( x ) sin ( y ) Comparing this to the given Step 2: ( A ) cos ( y ) − sin ( x ) sin ( y ) sin ( x ) cos ( y ) + cos ( x ) sin ( y ) We can see that A must be cos ( x ) .
Step 3 Analysis: Finding B Step 3 divides both the numerator and the denominator by a common factor. The denominator in Step 2 is cos ( x ) cos ( y ) − sin ( x ) sin ( y ) . In Step 3, the denominator is divided by cos ( x ) cos ( y ) to get 1 in the first term. Therefore, the numerator must also be divided by cos ( x ) cos ( y ) . However, in the numerator of Step 3, we see cos ( x ) ( B ) in the denominator of the fraction. This means that B must be cos ( y ) .
Step 4 and 5 Analysis: Finding C Step 4 simplifies the fractions by dividing each term by cos ( x ) cos ( y ) . Step 5 then converts these fractions into tangent functions. Specifically, c o s ( x ) s i n ( x ) = tan ( x ) and c o s ( y ) s i n ( y ) = tan ( y ) . Looking at the last term in the denominator of Step 4, we have c o s ( x ) c o s ( y ) s i n ( x ) s i n ( y ) = tan ( x ) tan ( y ) . Comparing this to Step 5, we see that C must be tan ( y ) .
Conclusion Therefore, we have found that A = cos ( x ) , B = cos ( y ) , and C = tan ( y ) .
Examples
The tangent sum identity is useful in physics, particularly in wave mechanics and optics, where angles of incidence and refraction are combined. For example, if you are analyzing the interference pattern of two light beams hitting a screen at different angles, you can use this identity to simplify calculations involving the combined angle and predict the intensity distribution on the screen. This helps in designing optical instruments and understanding wave phenomena.
In the derivation of the tangent sum identity, A is cos ( x ) , B is cos ( y ) , and C is tan ( y ) . This corresponds to replacing the placeholders in the derivation steps with the appropriate trigonometric identities. Thus, we conclude with A, B, and C filled in with their respective values.
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