1

Problem

Prove that \$4 (2 tan^(-1) (1/3) + tan^(-1) (1/7)) = (pi)\$.

2

Step

The given function is

\$4 (2 tan^(-1)(1/3) + tan^(-1) (1/7)) = (pi)\$

3

Step

Let’s start by letting:

\$α = tan^(-1) (1/3)\$ and \$β = tan^(-1) (1/7)\$

4

Formula

Since \$α = tan^(-1)(1/3​)\$, we have \$tanα =1/3​\$.

5

Hint

The given equation can be rewritten using α and β: 4(2α + β) = π.

6

Formula

Calculate tan⁡(2α) using the double-angle formula for tangent: \$tan⁡(2α) = tan⁡(2α) / (1- tan⁡^2(α))\$.

7

Step

Plug the values in the formula and then simplify it:

\$tan⁡(2α) = (2 \times 1/(3))/(1 - (1/3)^2)\$

\$ tan⁡(2α) = (2/(3))/(1- 1/(9))\$

\$tan⁡(2α) = (2/3)/(8/(9))\$

\$tan⁡(2α) = 3/4\$

8

Formula

The tangent of a sum of two angles is given by: \$tan(A + B) = (tanA + tanB)/(1 - tanA tanB)\$.

9

Formula

The formula is rewritten as calculate tan(2α+β) using the tangent addition: \$tan(2α + β) = (tan(2α) + tanβ)/(1 - tan(2α) tanβ)\$.

10

Step

Plug the values in the formula and then simplify it:

\$tan(2α + β) = (3/(4) + 1/(7))/(1 - 3/(4) . 1/(7))\$

\$tan(2α + β) = ((21)/ (28) + 4/(28))/(1 - 3/(28))\$

\$tan(2α + β) = (25/28)/(25/28)\$

\$tan(2α + β) = 1\$

11

Step

Since tan⁡(2α + β )= 1 , we know:

\$(2α + β) = tan^(-1) (1)\$

\$(2α + β) = (pi)/4\$

12

Step

Thus, substituting back, we get:

\$4(2α + β) = 4 \times (pi)/4 = (pi)\$

13

Solution

Therefore, the given equation: \$4 (2 tan^(-1)(1/3) + tan^(-1) (1/7)) = (pi)\$ is proven to be true.

14

Sumup

Please summarize Problem, Clue, Hint, Formula, Steps and Solution

Choices

15

Choice-A

This is incorrect because the statement we needed to prove does not result in the value 1; it results in π

Wrong 1

16

Choice-B

Option B is correct because the equation \$4(2tan^⁡(-1) (1/3) + tan⁡^(-1) (1/7)) = π\$ is proven to be true by using the formulas

Correct Proved

17

Choice-C

Wrong: Because the equation does not equal zero

Wrong Zero

18

Choice-D

The "Not proved" option is incorrect because we have shown that the original equation is true using trigonometric identities and properties of the arctangent function

Wrong Not proved

19

Answer

Option

B

20

Sumup

Please summarize choices