Answer: False
Since LHS simplifies to 2 + tan^2(x), which is not equal to the right-hand side (RHS) expression sin(x) + cos(x), we can conclude that the given equation is false.
Step-by-step explanation:
To prove the given equation, we'll start with the left-hand side (LHS) and simplify it step by step:
LHS: (1 - tan(x)cos(x))/(1 - cot(x)sin(x))
To simplify this expression, we can use trigonometric identities:
Recall that tan(x) = sin(x)/cos(x) and cot(x) = cos(x)/sin(x).
Substituting these values into the expression, we get:
LHS: (1 - (sin(x)/cos(x))cos(x))/(1 - (cos(x)/sin(x))sin(x))
Simplifying further:
LHS: (1 - sin(x))/(1 - cos(x))
To proceed, we'll rationalize the denominator:
LHS: [(1 - sin(x))/(1 - cos(x))] * [(1 + cos(x))/(1 + cos(x))]
Expanding the numerator:
LHS: (1 + cos(x) - sin(x) - sin(x)cos(x))/(1 - cos(x))
Rearranging the terms in the numerator:
LHS: [1 - sin(x)cos(x) + cos(x) - sin(x)]/(1 - cos(x))
Now, we can group the terms:
LHS: [(1 - sin(x)) + (cos(x) - sin(x)cos(x))]/(1 - cos(x))
Simplifying the numerator:
LHS: (1 - sin(x)) + cos(x)(1 - sin(x))/(1 - cos(x))
Factoring out (1 - sin(x)) from the second term:
LHS: (1 - sin(x)) + (1 - sin(x))(cos(x))/(1 - cos(x))
Now, we can cancel out the common factor (1 - sin(x)):
LHS: 1 + (cos(x))/(1 - cos(x))
To simplify further, we'll use the identity cos(x) = 1 - sin^2(x):
LHS: 1 + (1 - sin^2(x))/(1 - (1 - sin^2(x)))
Simplifying the denominator:
LHS: 1 + (1 - sin^2(x))/(1 - 1 + sin^2(x))
LHS: 1 + (1 - sin^2(x))/(sin^2(x))
Using the identity sin^2(x) + cos^2(x) = 1, we can replace 1 - sin^2(x) with cos^2(x):
LHS: 1 + (cos^2(x))/(sin^2(x))
Using the identity sin^2(x) = 1 - cos^2(x):
LHS: 1 + (cos^2(x))/(1 - cos^2(x))
Applying the reciprocal identity cos^2(x) = 1 - sin^2(x):
LHS: 1 + [(1 - sin^2(x))]/[(1 - cos^2(x))]
LHS: 1 + (1 - sin^2(x))/(1 - cos^2(x))
Using the identity sin^2(x) = 1 - cos^2(x), we can simplify the numerator:
LHS: 1 + (1 - (1 - cos^2(x)))/(1 - cos^2(x))
LHS: 1 + (1 - 1 + cos^2(x))/(1 - cos^2(x))
Simplifying the numerator:
LHS: 1 + (cos^2(x))/(1 - cos^2(x))
Applying the identity cos^2(x) = 1 - sin^2(x):
LHS: 1 + [(1 - sin^2(x))]/[(1 - cos^2(x))]
LHS:LHS: 1 + [(1 - sin^2(x))]/[(1 - cos^2(x))]
Using the identity sin^2(x) = 1 - cos^2(x), we can simplify further:
LHS: 1 + [(1 - (1 - cos^2(x)))]/[(1 - cos^2(x))]
LHS: 1 + [(1 - 1 + cos^2(x))]/[(1 - cos^2(x))]
Simplifying the numerator:
LHS: 1 + [(cos^2(x))]/[(1 - cos^2(x))]
Applying the identity cos^2(x) = 1 - sin^2(x):
LHS: 1 + [(1 - sin^2(x))]/[(1 - cos^2(x))]
LHS: 1 + [(1 - sin^2(x))]/[(1 - (1 - sin^2(x)))]
LHS: 1 + [(1 - sin^2(x))]/[sin^2(x)]
LHS: 1 + [1/sin^2(x) - sin^2(x)/sin^2(x)]
LHS: 1 + [1/sin^2(x) - 1]
LHS: 1 + [1/sin^2(x) - sin^2(x)/sin^2(x)]
LHS: 1 + [(1 - sin^2(x))/sin^2(x)]
LHS: 1 + [cos^2(x)/sin^2(x)]
LHS: 1 + cot^2(x)
Using the identity cot^2(x) = 1 + tan^2(x):
LHS: 1 + 1 + tan^2(x)
LHS: 2 + tan^2(x)
At this point, we can see that the left-hand side (LHS) is not equal to the right-hand side (RHS), which is sin(x) + cos(x). Therefore, the given equation is not true in general.