Question

Prove that : tan 1° tan 11° tan 21° tan 69° tan 79° tan 89° = 1.

Answer 1

Two angles are said to be complementary of their sum is equal to 90° .
θ & (90° - θ) are complementary angles.

SOLUTION:

GIVEN :
tan 1° tan 11° tan 21° tan 69° tan 79° tan 89°

LHS = tan 1° tan 11° tan 21° tan 69° tan 79° tan 89°
= tan(90° - 89°) tan(90° - 79°)  tan(90° - 69°) ….. tan 69° tan 79° tan 89°

[tan θ = cot(90 - θ )]

= cot 89°  cot 79° cot 69°….  tan 69° tan 79° tan 89°   
= (cot 89° tan 89°)  (cot 79° tan 79°) (cot 69° tan 69°)
= 1 × 1 × 1 = 1

[ cot θ ×  tan θ = 1]

Hence, cot 89°  cot 79° cot 69°….  tan 69° tan 79° tan 89° = 1

HOPE THIS WILL HELP YOU...

Answer 2

LHS
=tan (90°-89°) tan(90°-79°) tan(90°-69°) tan69° tan79° tan 89°
=(cot 89° tan 89° ) (cot 79° tan 79°) (cot 69° tan 69°)
=1×1×1=1 (tan thetha × cot thetha =1)=RHS
Hence proved.☺️☺️☺️☺️☺️☺️☺️