Question

The value of
`\tan 1^(\circ)\tan 2^(\circ)+\tan 2^(\circ)\tan 3^(\circ)+\tan 3^(\circ)\tan 4^(\circ)+\cdots+\tan 88^(\circ)\tan 89^(\circ)` can be expressed in the form
`\cot^2 1^(\circ)-n`. What is the value of
`n`?

Answer 1

n = 89

Explanation:

`\boxed{\begin{minipage}{5 cm}\underline{Tan double angle identity}\\\\$\tan (A - B)=(\tan A - \tan B)/(1 + \tan A \tan B)$\\\end{minipage}}`

Rewrite the tan double angle identity to isolate tanAtanB:

`\implies 1 +\tan A \tan B=(\tan A- \tan B)/(\tan (A - B))`

`\implies \tan A \tan B=(\tan A- \tan B)/(\tan (A - B))-1`

Therefore:

`\implies \tan 1^(\circ) \tan 2^(\circ)+\tan2^(\circ)\tan3^(\circ)+...+\tan88^(\circ)\tan89^(\circ)`

`\implies \left((\tan 1^(\circ)- \tan 2^(\circ))/(\tan (1 - 2)^(\circ))-1\right)+\left((\tan 2^(\circ)- \tan 3^(\circ))/(\tan (2 - 3)^(\circ))-1\right)+...+\left((\tan 88^(\circ)- \tan 89^(\circ))/(\tan (88 - 89)^(\circ))-1\right)`

`\implies (\tan 1^(\circ)- \tan 2^(\circ))/(\tan (-1)^(\circ))-1+(\tan 2^(\circ)- \tan 3^(\circ))/(\tan (-1)^(\circ))-1+...+(\tan 88^(\circ)- \tan 89^(\circ))/(\tan (-1)^(\circ))-1`

`\implies (\tan 1^(\circ)- \tan 2^(\circ))/(\tan (-1)^(\circ))+(\tan 2^(\circ)- \tan 3^(\circ))/(\tan (-1)^(\circ))+...+(\tan 88^(\circ)- \tan 89^(\circ))/(\tan (-1)^(\circ))-88`

`\implies (\tan 1^(\circ)- \tan 2^(\circ)+\tan 2^(\circ)- \tan 3^(\circ)+...+\tan 88^(\circ)- \tan 89^(\circ))/(\tan (-1)^(\circ))-88`

`\implies (\tan 1^(\circ)- \tan 89^(\circ))/(\tan (-1)^(\circ))-88`

`\implies (\tan 1^(\circ))/(\tan (-1)^(\circ))-( \tan 89^(\circ))/(\tan (-1)^(\circ))-88`

`\textsf{Apply the identity}\quad \boxed{ \tan(-x)=-\tan(x)}:`

`\implies (\tan 1^(\circ))/(-\tan 1^(\circ))+( \tan (90-1)^(\circ))/(\tan 1^(\circ))-88`

`\implies -1+( \tan (90-1)^(\circ))/(\tan 1^(\circ))-88`

`\implies ( \tan (90-1)^(\circ))/(\tan 1^(\circ))-89`

`\textsf{Apply the identity}\quad \boxed{\tan(90-x)=\cot(x)}:`

`\implies (\cot1^(\circ))/(\tan1^(\circ))-89`

`\implies \cot 1^(\circ) \cdot (1)/(\tan1^(\circ))-89`

`\textsf{Apply the identity}\quad \boxed{(1)/(\tan(x))=\cot(x)}:`

`\implies \cot 1^(\circ) \cdot \cot 1^(\circ)-89`

`\implies \cot^21^(\circ)-89`

Therefore, n = 89.

Answer 2

• n = 89

Explanation:

Use of identities

• tan (x - y) = (tan x - tan y)/(tan x tan y + 1)
• 1/tan x = cot x
• tan (90 - x) = cot x
• tan (- x) = - tan x

Convert the first one above:

• tan x tan y = (tan x - tan y) / tan (x - y) - 1

Apply this to each term of the given expression to get:

• tan1°tan2° + tan2°tan3° + ... + tan88°tan89° =
• (tan1° - tan2°)/(tan-1°) - 1 + (tan2° - tan3°)/(tan-1°) - 1 + ... + (tan88° - tan89°)/(tan-1°) - 1 =
• (tan1° - tan2° + tan2° - tan3° + ... + tan88° - tan89°)/(tan-1°) - 88 =
• (tan1° - tan89°)/(tan-1°) - 88 =
• tan1°/ (tan-1°) - tan89°/tan-1° - 88 =
• - 1 - cot1*(- cot1°) - 88 =
• cot² 1° - 89

The value of n is 89.