Verify the identity algebraically.
TAN X + COT Y/TAN X COT Y= TAN Y + COT X
Rewrite it as
(tanx + coty)/(tanx coty) = tany + cotx
LS = (sinx/cox + cosy/siny)/((sinx/cosx)(cosy/siny))
= [(sinxsiny + cosxcosy)/(cosxsiny) ] / [(sinxcosy)/(cosxsiny) ]
= (sinxsiny + cosxcosy)/(sinxcoy) (after multiplying by the reciprocal)
= sinxsiny/(sinxcosy) + cosycosx/)sinxcosy)
=siny/cosy + cosx/sinx
= tany + cotx
= RS
To verify the given identity algebraically, we'll simplify both sides of the equation.
Starting with the left side of the equation:
TAN X + COT Y / TAN X COT Y
To simplify, we can find a common denominator for the two terms in the numerator, which is TAN X COT Y. This gives us:
(TAN X * TAN X COT Y + COT Y) / (TAN X COT Y)
Now, using the fact that TAN X * COT Y = 1, we can substitute this value into the numerator:
((1) + COT Y) / (TAN X COT Y)
Now, we can simplify the numerator:
(1 + COT Y) / (TAN X COT Y)
Let's simplify the right side of the equation:
TAN Y + COT X
Now that we have simplified both sides of the equation, we can compare them:
(1 + COT Y) / (TAN X COT Y) = TAN Y + COT X
Thus, we have verified the identity algebraically.
To verify the given identity algebraically, we need to simplify both sides of the equation separately and show that they are equal. Let's start by simplifying the left-hand side (LHS) of the equation:
LHS: (TAN X + COT Y) / (TAN X * COT Y)
To simplify this expression, we need to use the trigonometric identities of tangent (TAN) and cotangent (COT). The identity we can apply here is:
TAN A + COT B = (SIN A / COS A) + (COS B / SIN B)
= (SIN^2 A + COS^2 B) / (SIN A * COS A)
Applying this identity to the LHS, we have:
LHS = [(SIN X / COS X) + (COS Y / SIN Y)] / (SIN X * COS Y)
To simplify further, we need to rationalize the denominator by multiplying both the numerator and denominator by (COS X * SIN Y):
LHS = [(SIN X * SIN Y * COS Y + COS X * COS Y * COS X) / (COS X * SIN Y * SIN X * COS Y)
Simplifying the numerator,
LHS = [SIN X * SIN Y * COS Y + COS X * COS Y * COS X] / (COS X * SIN Y * SIN X * COS Y)
= [SIN X * SIN Y * COS Y + COS^2 X * COS Y] / (COS X * SIN Y * SIN X * COS Y)
Factoring out a COS Y from the numerator,
LHS = COS Y * [SIN X * SIN Y + COS X * COS X] / (COS X * SIN Y * SIN X * COS Y)
Simplifying the numerator further,
LHS = COS Y * [SIN X * SIN Y + COS^2 X] / (COS X * SIN Y * SIN X * COS Y)
= COS Y * [1] / (COS X * SIN Y * SIN X * COS Y)
= COS Y / (COS X * SIN Y * SIN X * COS Y)
= 1 / (COS X * SIN X)
Now, let's simplify the right-hand side (RHS) of the equation:
RHS = TAN Y + COT X
Using the identity TAN A + COT B = (SIN A / COS A) + (COS B / SIN B), we can express the RHS as:
RHS = (SIN Y / COS Y) + (COS X / SIN X)
To simplify this further, we need to find a common denominator:
RHS = (SIN Y * SIN X + COS X * COS Y) / (COS Y * SIN X)
Comparing the RHS to the simplified LHS, we can see that they are both equal to 1 / (COS X * SIN X). Hence, the identity is verified algebraically.