prove that the following are identity
1. (cotA+cosecA-1)/(cotA-cosecA+1)=cosecA+cotA
2. (sinA-cosA+1)/(sinA+cosA-1)=
1/(secA-tanA)
To prove that the given expressions are identities, we need to simplify both sides of the equation until they are equivalent to each other. Let's start with the first expression:
1. (cotA + cosecA - 1) / (cotA - cosecA + 1) = cosecA + cotA
To simplify the left-hand side (LHS):
LHS = (cotA + cosecA - 1) / (cotA - cosecA + 1)
= [(cosA/sinA) + (1/sinA) - 1] / [(cosA/sinA) - (1/sinA) + 1] [Using the definitions of cotA and cosecA]
= [(cosA + 1 - sinA) / sinA] / [(cosA - 1 + sinA) / sinA]
= (cosA + 1 - sinA) / (cosA - 1 + sinA) [Multiplying both the numerator and denominator by sinA]
Now, let's simplify the right-hand side (RHS):
RHS = cosecA + cotA
= 1/sinA + cosA/sinA
= (cosA + 1 - sinA) / sinA [Adding fractions with the same denominator]
Comparing the simplified LHS and RHS, we can see that they are equal. Thus, we have proven the identity.
Now, let's move on to the second expression:
2. (sinA - cosA + 1) / (sinA + cosA - 1) = 1 / (secA - tanA)
To simplify the left-hand side (LHS):
LHS = (sinA - cosA + 1) / (sinA + cosA - 1)
To simplify the right-hand side (RHS):
RHS = 1 / (secA - tanA)
To proceed, we need to find a common denominator for sinA - cosA + 1 and sinA + cosA - 1.
Let's multiply the numerator and denominator of the LHS by (sinA + cosA + 1):
LHS = [(sinA - cosA + 1) / (sinA + cosA - 1)] * [(sinA + cosA + 1) / (sinA + cosA + 1)]
= (sinA^2 + sinA * cosA + sinA - sinA * cosA - cosA^2 - cosA + sinA + cosA + 1) / (sinA^2 + sinA * cosA + sinA - sinA * cosA + cosA^2 + cosA - sinA - cosA - 1)
= [sinA^2 - cosA^2 + 2(sinA - cosA) + 1] / [sinA^2 + cosA^2 - 2(sinA - cosA) - 1]
Now, let's simplify further:
LHS = [sinA^2 - cosA^2 + 2(sinA - cosA) + 1] / [sinA^2 + cosA^2 - 2(sinA - cosA) - 1]
= [(sinA + cosA)(sinA - cosA) + 2(sinA - cosA) + 1] / [(sinA + cosA)(sinA - cosA) - 2(sinA - cosA) - 1]
= [(sinA - cosA + 1)(sinA + cosA + 2)] / [(sinA - cosA + 1)(sinA + cosA - 2)]
Now, cancel out the common factors (sinA - cosA + 1):
LHS = (sinA + cosA + 2) / (sinA + cosA - 2)
This simplification matches the right-hand side (RHS) of the equation. Thus, the second expression is proven to be an identity.