Find the of $tan 1^{0} tan 2^{0} tan 3^{0} . . . . . . tan 45^{0} . . . . tan 80^{0} tan 89^{0}$

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Solution

The given expression can be written as $(tan 1^{0} tan 89^{0})$ $(tan 2^{0} tan 88^{0})$ .... $(tan 44^{0} tan 46^{0})$ $tan 45^{0}$
Now $tan 89^{0} = cot 1^{0}, tan 88^{0} = cot 2^{0}$ etc.
and $tan 45^{0} = 1$
$∴ (tan 1^{0} cot 1^{0}) . (tan 2^{0} cot 2^{0}) . . . . . (tan 44^{0} cot 44^{0}) \times 1 = 1 \times 1 \times 1........ \times 1 = 1$

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