The value of (tan$1^\circ$ tan$2^\circ$ tan$3^\circ$ ... tan$89^\circ$) is ______.

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The value of (tan$1^\circ$ tan$2^\circ$ tan$3^\circ$ ... tan$89^\circ$) is 1. Explanation: tan $1^\circ$.tan $2^\circ$.tan $3^\circ$ ...... tan $89^\circ$ = tan $1^\circ$.tan $2^\circ$.tan $3^\circ$...tan $43^\circ$.tan $44^\circ$.tan $45^\circ$.tan $46^\circ$.tan $47^\circ$...tan $87^\circ$.tan $88^\circ$.tan $89^\circ$ Since, tan $45^\circ$ = 1, = tan $1^\circ$.tan $2^\circ$.tan $3^\circ$...tan $43^\circ$.tan $44^\circ$.1.tan $46^\circ$.tan $47^\circ$...tan $87^\circ$.tan $88^\circ$.tan $89^\circ$ = tan $1^\circ$.tan $2^\circ$.tan $3^\circ$…tan $43^\circ$.tan $44^\circ$.1.tan($90^\circ$ – $44^\circ$).tan($90^\circ$ – $43^\circ$) ...tan($90^\circ$ – $3^\circ$).tan($90^\circ$ – $2^\circ$).tan($90^\circ$ – $1^\circ$) Since, tan($90^\circ$ – $\theta$) = cot $\theta$, = tan $1^\circ$.tan $2^\circ$.tan $3^\circ$...tan $43^\circ$.tan $44^\circ$.1.cot $44^\circ$.cot $43^\circ$...cot $3^\circ$.cot $2^\circ$.cot $1^\circ$ Since, tan $\theta$ = (1/cot $\theta$) = `tan1^circ * tan2^circ * tan$3^\circ$...tan43^circ * tan44^circ * 1 * (1/tan 44^circ). (1/tan 43^circ) ... (1/tan 3^circ) * (1/tan 2^circ) * (1/tan 1^circ) = (tan 1^circ xx 1/tan1^circ) * (tan 2^circ xx 1/tan 2^circ) ... (tan 44^circ xx 1/tan 44^circ)` = 1 Hence, tan $1^\circ$.tan $2^\circ$.tan $3^\circ$ ...... tan $89^\circ$ = 1

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