Question

Without using trigonometric tables prove that : tan 1 tan 11 tan 21 tan 69 tan 79 tan 89=1

Answer 1

tan1°tan11°tan21°tan69°tan79°tan89°
=tan1°tan11°tan21°tan(90°-21°)tan(90°-11°)tan(90°-1°)
=tan1°tan11°tan21°cot21°cot11°cot1°
=tan1°tan11°tan21°(1/tan21°)(1/tan11°)(1/tan1°)
=1 (Proved)

Answer 2

$\tan 1 \tan 11 \tan 21 \tan 69 \tan 79 \tan 89=1,$ proved.

Step-by-step explanation:

Prove that,  $\tan 1 \tan 11 \tan 21 \tan 69 \tan 79 \tan 89=1$.

L.H.S.$=\tan 1 \tan 11 \tan 21 \tan 69 \tan 79 \tan 89$

$=\tan 1 \tan 11 \tan 21 \tan (90-21) \tan (90-11) \tan (90-1)$

Since, $\cot A=\tan (90-A)$

$=\tan 1.\tan 11.\tan 21.\cot 21.\cot 11.\cot 1$

$=(\tan 1.\cot 1).(\tan 11.\cot 11).(\tan 21.\cot 21)$

Using trigonometric formula,

$\tan A.\cot A=1$

$=(1).(1).(1)$

= 1

= R.H.S., proved,

Hence, $\tan 1 \tan 11 \tan 21 \tan 69 \tan 79 \tan 89=1$, proved.