Question

Evaluate without using trigonometric tables :

tan 20 .tan 40 .tan 50 .tan 70.​

Answer 1

Required Answer:-

We have to evaluate:

• $\rm{ \tan(20) . \tan(40) . \tan(50) . \tan(70) }$

By the concept of complementary angles:

• sin x = cos (90-x)
• tan x = cot (90-x)
• sec x = cosec(90-x) and vice versa....

Then,

Assuming they are in degrees (°)

➻ tan 20. tan 40. tan (90 - 40). tan (90 - 20)

➻ tan 20. tan 40. cot 40. cot 20

➻ tan 20. cot 20. tan 40. cot 40

➻ 1 × 1 = 1 (A)

(Because we know that, tan x. cot x = 1)

Note:

Complementary angles are the angles that add upto 90°. Like in the above case,

• 20° and 70° are complementary angles.
• 40° and 50° are complementary angles.

Answer 2

$\huge\fbox \red{✔Que} {\colorbox{crimson}{st}}\fbox\red{ion✔}$

Evaluate without using trigonometric tables :

tan 20 .tan 40 .tan 50 .tan 70.

$\huge\fbox \red{✔AN} {\colorbox{crimson}{SW}}\fbox\red{ER✔}$

GIVEN:-

tan20 ° tan40 ° tan50 ° tan70 °

TO FIND:-

the value of tan20 ° tan40 ° tan50 ° tan70 ° = ?

SOLUTION:-

∴ tan20 ° tan40 ° tan50 ° tan70 °

= tan20 ° tan40 ° tan(90−40) tan(90−20)

Using the trigonometric identity,

cotA = tan(90−A)

= tan20 ° tan40 ° cot40 ° cot20 °

= ( tan20 °cot20 °) × ( tan40 ° cot40 °)

Using the trigonometric identity,

tanA ° cotA = 1

= 1 × 1

= 1

∴ tan20 ° tan40 ° tan50 ° tan70 ° = 1

• I Hope it's Helpful My Friend.