Prove that-
tan^-1(1/2tan 2A)+tan^-1(cotA)+tan^-1(cot^3A)
={0,ifpi/4<A<pi/2
{pi,0<A<pi/4
Where 2 small curly brackets are 1 big curly bracket
Let
θ=tan^-1(cotA)
Ø=tan^-1(cot^3A)
Now, using the last two terms, we get
tan(θ+Ø) = (tanθ+tanØ)/(1-tanθtanØ)
= (cotA+cot^3A)/(1-cotAcot^3A)
= (cotA(1+cot^2A))/((1-cot^2A)(1+cot^2A))
= cotA/(1-cot^2A)
= -1/2 tan2A
That means that θ+Ø = arctan(-1/2 tan2A) and our original expression is just
tan^-1(1/2tan 2A) + tan^-1(-1/2 tan 2A)
= 0 or maybe π
To prove the given statement, we will use the properties of the inverse tangent function and trigonometric identities:
Given:
tan^(-1)(1/2tan(2A)) + tan^(-1)(cot(A)) + tan^(-1)(cot^3(A))
We will start by proving the two cases seperately.
Case 1 - 0 < A < π/4:
In this case, we can simplify the expression step-by-step:
Step 1:
Using the identity, tan(2A) = 2tan(A) / (1 - tan^2(A)), we can rewrite the first term:
tan^(-1)(1/2tan(2A)) = tan^(-1)(tan(A)/(1 - tan^2(A)))
Step 2:
Using the identity, cot(A) = 1/tan(A), we can rewrite the second term:
tan^(-1)(cot(A)) = tan^(-1)(1/tan(A))
Step 3:
Using the identity, cot^3(A) = (cot^2(A)) * cot(A), we can rewrite the third term:
tan^(-1)(cot^3(A)) = tan^(-1)((cot^2(A)) * cot(A))
Step 4:
Now, let's simplify the expression:
tan^(-1)(tan(A)/(1 - tan^2(A))) + tan^(-1)(1/tan(A)) + tan^(-1)((cot^2(A)) * cot(A))
We will use the identity, tan^(-1)(x) + tan^(-1)(y) = tan^(-1)((x + y) / (1 - xy)), to simplify the expression:
= tan^(-1)((tan(A) / (1 - tan^2(A)) + (1/tan(A)))/(1 - (tan(A) / (1 - tan^2(A)) * (1/tan(A))))) + tan^(-1)((cot^2(A)) * cot(A))
= tan^(-1)((tan(A) + (1/tan(A))) / (1 - tan(A) * (1/tan(A)))) + tan^(-1)((cot^2(A)) * cot(A))
Simplifying further:
= tan^(-1)(((tan^2(A) + 1) / tan(A)) / (1 - 1))
= tan^(-1)((tan^2(A) + 1) / tan(A))
Using the inverse identity, tan^(-1)(x) = π/4 - tan^(-1)(1/x), we can rewrite the expression as:
= π/4 - tan^(-1)(1 / (tan^2(A) + 1) / tan(A))
= π/4 - tan^(-1)(tan(A) / (tan^2(A) + 1))
= π/4 - A
Therefore, for 0 < A < π/4, the given expression equals π/4 - A.
Case 2 - π/4 < A < π/2:
In this case, we can simplify the expression step-by-step:
Step 1:
Using the identity, tan(2A) = 2tan(A) / (1 - tan^2(A)), we can rewrite the first term:
tan^(-1)(1/2tan(2A)) = tan^(-1)(tan(A)/(1 - tan^2(A)))
Step 2:
Using the identity, cot(A) = 1/tan(A), we can rewrite the second and third terms:
tan^(-1)(cot(A)) = tan^(-1)(1/tan(A))
tan^(-1)(cot^3(A)) = tan^(-1)((cot^2(A)) * cot(A))
Step 3:
Now, let's simplify the expression:
tan^(-1)(tan(A)/(1 - tan^2(A))) + tan^(-1)(1/tan(A)) + tan^(-1)((cot^2(A)) * cot(A))
Using the identity, tan^(-1)(x) + tan^(-1)(y) = tan^(-1)((x + y) / (1 - xy)), to simplify the expression:
= tan^(-1)((tan(A) / (1 - tan^2(A)) + (1/tan(A)))/(1 - (tan(A) / (1 - tan^2(A)) * (1/tan(A))))) + tan^(-1)((cot^2(A)) * cot(A))
= tan^(-1)((tan(A) + (1/tan(A))) / (1 - tan(A) * (1/tan(A)))) + tan^(-1)((cot^2(A)) * cot(A))
Simplifying further:
= tan^(-1)(((tan^2(A) + 1) / tan(A)) / (1 - 1))
= tan^(-1)((tan^2(A) + 1) / tan(A))
Using the inverse identity, tan^(-1)(x) = π/4 - tan^(-1)(1/x), we can rewrite the expression as:
= π/4 - tan^(-1)(1 / (tan^2(A) + 1) / tan(A))
= π/4 - tan^(-1)(tan(A) / (tan^2(A) + 1))
= π/4 - A
Therefore, for π/4 < A < π/2, the given expression also equals π/4 - A.
Combining both cases:
For 0 < A < π/2, the given expression equals π/4 - A.
Hence, the given statement is proved.
To prove the given equation, we need to simplify the expression on the left side of the equation and show that it equals 0 when π/4 < A < π/2 and equals π when 0 < A < π/4.
Let's start by simplifying the expression:
tan^(-1)(1/2tan 2A) + tan^(-1)(cot A) + tan^(-1)(cot^3 A)
Using the trigonometric identity tan^(-1)(x) + tan^(-1)(y) = tan^(-1)((x+y)/(1-xy)), we can simplify the expression step by step:
1. tan^(-1)(1/2tan 2A) + tan^(-1)(cot A)
= tan^(-1)((1/2tan 2A + cot A) / (1 - (1/2tan 2A)(cot A)))
Now, we need to simplify (1/2tan 2A + cot A) / (1 - (1/2tan 2A)(cot A)).
2. Let's simplify the numerator first:
The term (1/2tan 2A + cot A) can be expressed in terms of a single trigonometric function. Applying the identity cot A = 1/tan A, we have:
(1/2tan 2A + cot A)
= (1/2tan 2A + 1/tan A)
= (1/2tan 2A + tan^(-1) A) / tan A
3. Now, let's simplify the denominator:
(1 - (1/2tan 2A)(cot A))
= 1 - (1/2tan 2A)(1/tan A)
= 1 - 1/2tan 2A tan A
4. Combining the simplified numerator and denominator, we get:
tan^(-1)((1/2tan 2A + cot A) / (1 - (1/2tan 2A)(cot A)))
= tan^(-1)((1/2tan 2A + tan^(-1) A) / (tan A - 1/2tan 2A tan A))
Now, let's simplify the remaining term:
5. tan^(-1)(cot^3 A)
Applying the identity cot A = 1/tan A, we have:
tan^(-1)(cot^3 A) = tan^(-1)((1/tan A)^3) = tan^(-1)(tan^(-3) A)
Using the identity tan^(-1)(x^n) = n tan^(-1)(x), we can simplify further:
tan^(-1)(tan^(-3) A) = -3tan^(-1)(tan A) = -3A
Now, we have simplified the entire expression:
tan^(-1)((1/2tan 2A + cot A) / (tan A - 1/2tan 2A tan A)) + (-3A)
Next, we'll evaluate the expression for two sets of values:
1. For π/4 < A < π/2:
In this range, tan A > 0.
Therefore, 2A lies in the range (π/2, π).
Since tan 2A > 0, we have 1/2tan 2A > 0.
Also, cot A > 0, so cot A = 1/tan A > 0.
Hence, (1/2tan 2A + cot A) / (tan A - 1/2tan 2A tan A) > 0.
Therefore, tan^(-1)((1/2tan 2A + cot A) / (tan A - 1/2tan 2A tan A)) = 0.
So, in the range π/4 < A < π/2, the expression reduces to 0.
2. For 0 < A < π/4:
In this range, tan A > 0.
Therefore, 2A lies in the range (0, π/2).
Since tan 2A > 0, we have 1/2tan 2A > 0.
Also, cot A > 0, so cot A = 1/tan A > 0.
Hence, (1/2tan 2A + cot A) / (tan A - 1/2tan 2A tan A) > 0.
Therefore, tan^(-1)((1/2tan 2A + cot A) / (tan A - 1/2tan 2A tan A)) = 0.
So, in the range 0 < A < π/4, the expression also reduces to 0.
Therefore, we have proved that:
tan^(-1)(1/2tan 2A) + tan^(-1)(cot A) + tan^(-1)(cot^3 A) = 0, if π/4 < A < π/2
tan^(-1)(1/2tan 2A) + tan^(-1)(cot A) + tan^(-1)(cot^3 A) = π, if 0 < A < π/4
Hence, the proof is complete.