(TanA/1-cotA)+CotA/1-TanA
To simplify the given expression, we can rewrite the terms in terms of sine and cosine:
TanA/1-cotA = sinA/cosA / (1 - cosA/sinA) = sinA/cosA * (sinA/sinA) / (sinA - cosA)/sinA = sin^2A / (cosA(sinA-cosA))
CotA/1-TanA = cosA/sinA / (1 - sinA/cosA) = cosA/sinA * (cosA/cosA) / (cosA -sinA)/cosA = cos^2A / (sinA(cosA-sinA))
So, the given expression simplifies to:
(sin^2A / (cosA(sinA-cosA))) + (cos^2A / (sinA(cosA-sinA)))
Now, we can find the common denominator and add the fractions:
(sin^2A * cosA) / (cosA(sinA-cosA)) + (cos^2A * sinA) / (sinA(cosA-sinA))
= (sin^2A * cosA + cos^2A * sinA) / (cosA(sinA-cosA))
Since sin^2A + cos^2A = 1, we can simplify the numerator:
((1 - cos^2A) * cosA + cos^2A * sinA) / (cosA(sinA-cosA))
= (cosA - cos^3A + cos^2A * sinA) / (cosA(sinA-cosA))
= (cosA(1 - cos^2A + cosA * sinA)) / (cosA(sinA-cosA))
= (1 - cos^2A + cosA * sinA) / (sinA-cosA)
= (1 + cosA * sinA) / (sinA-cosA)
TanA/(1-cotA) + CotA/(1-TanA)
= tanA + cotA + 1
There are many ways of simplifying/rewriting this.
Here is another way of simplifying the expression:
To simplify the given expression, we can rewrite the terms in terms of sine and cosine:
TanA/(1-cotA) = sinA/cosA / (1 - cosA/sinA) = sinA/cosA * (sinA/sinA) / (sinA - cosA)/sinA = sin^2A / (cosA(sinA-cosA))
CotA/(1-TanA) = cosA/sinA / (1 - sinA/cosA) = cosA/sinA * (cosA/cosA) / (cosA -sinA)/cosA = cos^2A / (sinA(cosA-sinA))
So, the given expression simplifies to:
(sin^2A / (cosA(sinA-cosA))) + (cos^2A / (sinA(cosA-sinA)))
Now, we can find the common denominator and add the fractions:
(sin^2A * cosA) / (cosA(sinA-cosA)) + (cos^2A * sinA) / (sinA(cosA-sinA))
= (sin^2A * cosA + cos^2A * sinA) / (cosA(sinA-cosA))
Since sin^2A + cos^2A = 1, we can simplify the numerator:
((1 - cos^2A) * cosA + cos^2A * sinA) / (cosA(sinA-cosA))
= (cosA - cos^3A + cos^2A * sinA) / (cosA(sinA-cosA))
= (cosA(1 - cos^2A + cosA * sinA)) / (cosA(sinA-cosA))
= (1 - cos^2A + cosA * sinA) / (sinA-cosA)
= (1 + cosA * sinA) / (sinA-cosA)
Therefore, TanA/(1-cotA) + CotA/(1-TanA) simplifies to (1 + cosA * sinA) / (sinA-cosA).
To simplify the expression (TanA/1-cotA) + (CotA/1-TanA), we can first find the common denominator and then combine the terms.
Step 1: Find the common denominator
For the first term, the denominator is (1 - cotA), and for the second term, the denominator is (1 - TanA). To find the common denominator, we need to multiply these two denominators together:
(1 - cotA) * (1 - TanA) = (1 - cotA - TanA + cotA * TanA)
Step 2: Simplify each term
The first term (TanA/1-cotA) can be simplified by multiplying the numerator and denominator by (1 + cotA):
(TanA * (1 + cotA)) / ((1 - cotA) * (1 + cotA)) = (TanA + TanA * cotA) / (1 - cotA^2)
The second term (CotA/1-TanA) can be simplified by multiplying the numerator and denominator by (1 + TanA):
(CotA * (1 + TanA)) / ((1 - TanA) * (1 + TanA)) = (CotA + CotA * TanA) / (1 - TanA^2)
Step 3: Combine the terms
Now that each term is simplified, we can add them together:
(TanA + TanA * cotA) / (1 - cotA^2) + (CotA + CotA * TanA) / (1 - TanA^2)
Step 4: Simplify the common denominators
The two denominators, 1 - cotA^2 and 1 - TanA^2, can be further simplified to 1 - cos^2(A) and 1 - sin^2(A) respectively, using the trigonometric identities.
Step 5: Apply trigonometric identities
1 - cos^2(A) can be simplified to sin^2(A), and 1 - sin^2(A) can be simplified to cos^2(A).
Therefore, the expression becomes:
(TanA + TanA * cotA) / sin^2(A) + (CotA + CotA * TanA) / cos^2(A)
To simplify the expression (tanA/1-cotA) + (cotA/1-tanA), we need to manipulate the terms to a common denominator and combine them into a single fraction.
Let's start by working on the first term: (tanA/1-cotA).
First, we observe that cotA is equal to 1/tanA. Therefore, we can substitute 1/tanA for cotA in the expression:
(tanA/1-cotA) = (tanA/1-(1/tanA))
Next, we need to find a common denominator for 1 and (1/tanA). The common denominator is tanA. To achieve this, we multiply (1/tanA) by (tanA/tanA), which gives us:
(tanA/1-cotA) = (tanA/1-(1/tanA)) = (tanA/1-(1/tanA)) * (tanA/tanA)
= (tanA*tanA-1)/tanA = (tan^2(A) - 1)/tanA
Moving on to the second term: (cotA/1-tanA).
Similarly, we can substitute 1/tanA for cotA:
(cotA/1-tanA) = (1/tanA)/(1-tanA)
To combine the two terms into a single fraction, we need to find a common denominator for (tan^2(A) - 1)/tanA and (1/tanA)/(1-tanA).
The common denominator is tanA*(1-tanA).
Multiply the numerator and denominator of (tan^2(A) - 1)/tanA by (1-tanA) to have a common denominator:
((tan^2(A) - 1)/(tanA)) * ((1-tanA)/(1-tanA)) = (tan^2(A) - 1)*(1-tanA)/(tanA*(1-tanA))
Thus, the expression (tanA/1-cotA) + (cotA/1-tanA) simplifies to:
(tan^2(A) - 1)*(1-tanA)/(tanA*(1-tanA))
Now, notice that (1-tanA) can be factored out:
(tan^2(A) - 1)*(1-tanA)/(tanA*(1-tanA))
= (tan^2(A) - 1)*(1-tanA)/tanA
Further simplifying, we can factor out a (tan^2(A) - 1) common to both terms in the numerator:
(tan^2(A) - 1)*(1-tanA)/tanA
= ((tan(A) - 1)*(tan(A) + 1))*(1-tanA)/tanA
The final simplified form of the expression is:
((tan(A) - 1)*(tan(A) + 1))*(1-tanA)/tanA