(1+tan^2A)/(1+cot^2A)=((1-tanA)^2)/(1-cotA)^2
ok - I think the point of the exercise, though, is to prove it.
Stupid bot
Left Side:
(1+tan^2A)/(1+cot^2A)
= sec^2A/csc^2A
= tan^2A
Right Side:
(1-tanA)^2/(1-cotA)^2
= (1 - sinA/cosA)^2/(1 - cosA/sinA)^2
= ((cosA-sinA)/cosA)^2 / ((sinA-cosA)/sinA)^2
= sin^2A/cos^2A
= tan^2A
QED
The question probably was to "prove it", not to agree or disagree with the
statement.
LS = (1+tan^2A)/(1+cot^2A)
= sec^2 A/csc^2 A)
= sin^2 A/cos^2 A = tan^2 A
RS = ((1-tanA)^2)/(1-cotA)^2
= ( (1 - tanA)/(1 - cotA) )^2
= [ ( 1 - sinA/cosA)/(1 - cosA/sinA) ]^2
= [ ((cosA - sinA)/cosA)/((sinA - cosA)/sinA) ]^2
= [ - sinA/cosA]^2
= [-tanA]^2
= tan^2 A
= LS
To prove the equation:
(1 + tan^2A)/(1 + cot^2A) = ((1 - tanA)^2)/(1 - cotA)^2
We will start with the left-hand side of the equation and simplify it step by step using trigonometric identities:
Step 1: Start with the left-hand side of the equation:
(1 + tan^2A)/(1 + cot^2A)
Step 2: Rewrite the numerator and denominator using trigonometric identities:
(1 + sin^2A/cos^2A)/(1 + cos^2A/sin^2A)
Step 3: Combine the fractions in the numerator and denominator:
[(cos^2A + sin^2A)/(cos^2A)] / [(sin^2A + cos^2A) / (sin^2A)]
Step 4: Since (cos^2A + sin^2A) = 1, the equation simplifies to:
(1/cos^2A) / (1/sin^2A)
Step 5: Invert and multiply the denominator by its reciprocal to simplify the equation:
(1/cos^2A) * (sin^2A/1)
Step 6: Multiply the numerators and denominators:
(sin^2A)/(cos^2A)
Step 7: Apply the identity tan^2A = sin^2A/cos^2A to the equation:
tan^2A
Now, let's move on to the right-hand side of the equation:
Step 1: Start with the right-hand side of the equation:
((1 - tanA)^2)/(1 - cotA)^2
Step 2: Expand the squared terms:
(1 - 2tanA + tan^2A)/(1 - 2cotA + cot^2A)
Step 3: Rewrite the numerator and denominator using trigonometric identities:
(1 - 2sinA/cosA + sin^2A/cos^2A)/(1 - 2cosA/sinA + cos^2A/sin^2A)
Step 4: Combine the fractions in the numerator and denominator:
[(cos^2A - 2sinAcosA + sin^2A)/(cos^2A)] / [(sin^2A - 2cosAsinA + cos^2A) / (sin^2A)]
Step 5: Since (cos^2A - 2sinAcosA + sin^2A) = (cosA - sinA)^2, the equation simplifies to:
[(cosA - sinA)^2 / (cos^2A)] / [(sinA - cosA)^2 / (sin^2A)]
Step 6: Invert and multiply the denominator by its reciprocal to simplify the equation:
[(cosA - sinA)^2 / (cos^2A)] * [(sin^2A) / (sinA - cosA)^2]
Step 7: Multiply the numerators and denominators:
(sin^2A * (cosA - sinA)^2) / (cos^2A * (sinA - cosA)^2)
Step 8: Rewrite the squared terms:
(sin^2A * (cosA - sinA) * (cosA - sinA)) / (cos^2A * (sinA - cosA) * (sinA - cosA))
Step 9: Cancel out common factors:
(sin^2A * (cosA - sinA)) / (cos^2A * (sinA - cosA))
Step 10: Apply the identity tanA = sinA/cosA to the equation:
tanA * (1 - tanA) / ((1 - tanA) * (1 + tanA))
Step 11: Cancel out common factors:
tanA / (1 + tanA)
By comparing the left-hand side and the right-hand side of the equation, we see that they are equal (both are equal to tan^2A). Therefore, the equation is proved.