Verify the identity: sin(A+B)/sin(A-B) = tan(A)+tan(B)/tan(A)-tan(B) Explain steps
sin(a+b)/sin(a-b)
= (sina cosb + cosa sinb)/(sina cosb - cosa sinb)
now divide top and bottom by cosa cosb and you get
(sina/cosa + sinb/cosb)/(sina/cosa - sinb/cosb)
= (tana + tanb)/(tana - tanb)
No
What do you mean no.....................
What do you mean no..............................
Sure! Let's verify the given identity step by step:
1. Start with the left-hand side (LHS) of the equation: sin(A+B)/sin(A-B).
2. Recall the trigonometric identity for the sum of angles: sin(A+B) = sin(A)cos(B) + cos(A)sin(B).
3. Apply the identity to the numerator of the LHS: sin(A)cos(B) + cos(A)sin(B).
4. Next, consider the denominator of the LHS: sin(A-B).
5. Again, use the trigonometric identity for the difference of angles: sin(A-B) = sin(A)cos(B) - cos(A)sin(B).
6. Substitute the denominator in the LHS: sin(A+B) / (sin(A)cos(B) - cos(A)sin(B)).
7. Now, let's simplify the right-hand side (RHS): tan(A) + tan(B) / tan(A) - tan(B).
8. Recall the definitions of tan(A) and tan(B): tan(A) = sin(A)/cos(A) and tan(B) = sin(B)/cos(B).
9. Substitute the values in the RHS: (sin(A)/cos(A)) + (sin(B)/cos(B)) / (sin(A)/cos(A)) - (sin(B)/cos(B)).
10. Cross multiply the numerator and denominator of the RHS: (sin(A)cos(B) + cos(A)sin(B)) / (sin(A)cos(B) - cos(A)sin(B)).
11. Observe that the LHS and RHS are the same: (sin(A+B) / (sin(A)cos(B) - cos(A)sin(B)) = (sin(A+B) / (sin(A)cos(B) - cos(A)sin(B)).
Thus, we have verified the identity sin(A+B)/sin(A-B) = tan(A) + tan(B) / tan(A) - tan(B).
To verify the given identity:
Step 1: Start with the left-hand side (LHS) of the equation: sin(A+B)/sin(A-B).
Step 2: Use the trigonometric identity for the sine of the sum and the difference of angles:
sin(X+Y) = sin(X)cos(Y) + cos(X)sin(Y) and
sin(X-Y) = sin(X)cos(Y) - cos(X)sin(Y).
Applying these identities to the numerator and denominator of the LHS:
sin(A+B) = sin(A)cos(B) + cos(A)sin(B) and
sin(A-B) = sin(A)cos(B) - cos(A)sin(B).
Step 3: Substitute the above values back into the LHS:
(sin(A)cos(B) + cos(A)sin(B)) / (sin(A)cos(B) - cos(A)sin(B)).
Step 4: Simplify the numerator and denominator separately:
numerator: sin(A)cos(B) + cos(A)sin(B) = sin(A)cos(B) + sin(B)cos(A).
denominator: sin(A)cos(B) - cos(A)sin(B) = sin(A)cos(B) - sin(B)cos(A).
Step 5: Apply the commutative property of addition in the numerator and denominator:
numerator: sin(A)cos(B) + sin(B)cos(A) = cos(A)sin(B) + sin(A)cos(B).
denominator: sin(A)cos(B) - sin(B)cos(A) = cos(A)sin(B) - sin(B)cos(A).
Step 6: Factor out sin(A)cos(B) from both the numerator and denominator:
numerator: cos(A)sin(B) + sin(A)cos(B) = cos(A) sin(B) * (1 + cot(A)).
denominator: cos(A)sin(B) - sin(B)cos(A) = cos(A) sin(B) * (1 - cot(A)).
Step 7: Cancel out the common factor (cos(A)sin(B)):
(cos(A) sin(B) * (1 + cot(A))) / (cos(A) sin(B) * (1 - cot(A))).
Step 8: Remove the common factor from the numerator and denominator:
(1 + cot(A)) / (1 - cot(A)).
Step 9: Use the reciprocal identity for cotangent:
cot(A) = 1 / tan(A).
Step 10: Substitute back into the expression:
(1 + 1/tan(A)) / (1 - 1/tan(A)).
Step 11: Combine the fractions by finding a common denominator:
(tan(A) + 1) / (tan(A) - 1).
Step 12: The LHS is now equal to the right-hand side (RHS) of the equation. Thus, the identity is verified:
sin(A+B)/sin(A-B) = (tan(A) + tan(B))/(tan(A) - tan(B)).