Question

prove that tan 9 degree X tan 21 degree X tan 69 degree X and 81 degree equal to 1

Answer 1

tan9degreextan81degree×tan21degreextan69degree
tan9degree.cot9degree.tan21degree.cot21degree
1.1
=1

Answer 2

$\tan 9$° × $\tan 21$° × $\tan 69$° × $\tan 81$° = 1, proved.

Step-by-step explanation:

To prove that: $\tan 9$° × $\tan 21$° × $\tan 69$° × $\tan 81$° = 1.

L.H.S. = $\tan 9$° × $\tan 21$° × $\tan 69$° × $\tan 81$°

= $\tan 9$° × $\tan 21$° × $\tan (90-21)$° × $\tan (90-9)$°

Using the trigonometric identity,

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

= $\tan 9$° × $\tan 21$° × $\cot 21$° × $\cot 9$°

= ($\tan 9$° × $\cot 9$°) × ($\tan 21$° × $\cot 21$°)

Using the trigonometric identity,

$\tan A$ × $\cot A$ = 1

⇒ $\cot A$ = $\dfrac{1}{ \tan A}$

= ($\tan 9$° × $\dfrac{1}{\tan 9}$°) × ($\tan 21$° × $\dfrac{1}{\tan 21}$°)

= 1 × 1

= 1

= R.H.S., proved.

Thus, $\tan 9$° × $\tan 21$° × $\tan 69$° × $\tan 81$° = 1, proved.