Question

Evaluate the following without using a table:

2 tan 18° tan 72° − cot² 30° + cos 34° cosec 56°

Answer 1

Given:

2 tan 18° tan 72° − cot² 30° + cos 34° cosec 56°

Step 1: 

tan A ⋅ tan(90° − A) = 1 ⇒ tan 18° ⋅ tan 72° = 1

2 ⋅ tan 18° ⋅ tan 72° = 2 ⋅ 1 = 2

Step 2: 

cot 30° = `1/(tan 30°)`

= `(1/1)/sqrt3 = sqrt3 ⇒ cot^2 30° = (sqrt3)^2 = 3`

Step 3: 

cos34° ⋅ csc56°

cosec 56° = `1/(sin 56°) and sin 56° = cos(34°)`

cosec 56° = `1/(cos 34°) ⇒ cos 34° ⋅  "cosec"  56° = cos 34° ⋅ 1/cos 34° = 1`

2 tan 18° tan 72° − cot² 30° + cos 34° cosec 56°

= 2 − 3 + 1 = 0     ... [Putting all values together]