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
3 answers
TanA/(1-cotA) + CotA/(1-TanA)
= tanA + cotA + 1
There are many ways of simplifying/rewriting this.
= 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 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).
1 answer