Question

Evaluate the following without using a table:

3 tan 28° tan 62° − `(sec 36°)/("cosec" 54°) + (sin 28°)/(cos 62°)`

Answer 1

Given:

3 tan 28° tan 62° − `(sec 36°)/("cosec" 54°) + (sin 28°)/(cos 62°)`

Step 1:

Complementary angle identities:

tan(90° − θ) = cot θ

cos(90° − θ) = sin θ

sin(90° − θ) = cosec θ

sec(90° − θ) = cosec θ

cosec(90° − θ) = sec θ

Step 2: Simplify each term

3 tan 28° tan 62°      ...[First term]

tan 62° = cot 28° → tan 28° ⋅ cot 28°    

= 1 → 3 × 1 = 3   

`(sec 36°)/("cosec" 54°)`    ...[Second term]

cosec 54° = sec 36° → `(sec 36°)/(sec 36°)` = 1    

`(sin 28°)/(cos 62°)`     ...[Third term]

cos 62° = sin 28° → `(sin 28°)/(sin 28°) = 1`

= 3 − 1 + 1

= 3