Algebraically find the value of: 10 cot⁡(cot^(-1)⁡3+cot^(-1)⁡7+cot^(-1)⁡13+cot^(-1)⁡21 )

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http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibpi.html

and scroll down to Sec. 6.2

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To find the value of the expression 10cot(cot^(-1)3 + cot^(-1)7 + cot^(-1)13 + cot^(-1)21), we need to simplify it algebraically using the properties of trigonometric functions.

Step 1: Recall the relationship between the cotangent and its inverse function. If cot^(-1)x = y, then cot(y) = x.

Step 2: We will start by evaluating each nested cot^(-1) expression one at a time. Let's call the expression cot^(-1)3 as A, cot^(-1)7 as B, cot^(-1)13 as C, and cot^(-1)21 as D.

Step 3: To simplify cot(A + B + C + D), we need to find the cotangent of each individual angle and then sum them up. Let's calculate them one by one:

cot(A) = 3 (from the given cot^(-1)3)
cot(B) = 7 (from the given cot^(-1)7)
cot(C) = 13 (from the given cot^(-1)13)
cot(D) = 21 (from the given cot^(-1)21)

Step 4: Now we have cot(A), cot(B), cot(C), and cot(D). We can add them together:

cot(A + B + C + D) = cot(A) + cot(B) + cot(C) + cot(D)
= 3 + 7 + 13 + 21
= 44.

Step 5: Finally, we can substitute this value back into the original expression:

10cot(cot^(-1)3 + cot^(-1)7 + cot^(-1)13 + cot^(-1)21)
= 10cot(44)
= 10 * (1 / tan(44))
(Note: cot(x) is equal to 1 divided by tan(x)).
= 10 / tan(44)

Therefore, the value of the given expression is 10 / tan(44).