Question

Evaluate the following :

(sin²40° + sin²50°) / (cosec²70° - tan²20°) + 2 sec²58° - 2 tan 58° cot 32° - 4 tan 48° tan 23° tan 42° tan 67°.   

1. 1
2. 0
3. -2
4. -1

Answer 1

Correct Answer - Option 4 : -1

Given∶

A trigonometric expression (sin240° + sin250°) / (cosec270° - tan220°) + 2 sec258° - 2 tan 58° cot 32° - 4 tan 48° tan 23° tan 42° tan 67°.

Formula Used∶

Basics of trigonometry; Trigonometic Ratios of complementary angles.

Trigonometric Identities ; sec²θ - tan²θ = 1 ; cosec²θ - cot²θ = 1, sin²θ + cos²θ = 1

Calculation∶

 

sin240° + sin250°) / (cosec270° - tan220°) + 2 sec258° - 2 tan 58° cot 32° - 4 tan 48° tan 23° tan 42° tan 67°

⇒ [sin²40° + sin²(90° -  40°)] / [cosec²70°- tan²(90° - 70°)] + 2 [sec² 58° - tan 58° . cot (90° -  58°)] - 4(tan 48°  tan 42°) (tan 23° tan 67°)

⇒ (sin²40° + cos²40°) / (cosec²70° - cot²70°) + 2 [sec²58° - tan 58° . tan 58°] - 4[tan 48°. tan (90° - 48°)] [tan 23° . tan(90° - 23°]

⇒ 1/1 + 2 [sec²58° - tan²58°] - 4 [(tan 48°.cot 48°) (tan 23°. cot 23°)]

⇒ 1 + 2(1) - 4 (1) (1)

⇒ 3 - 4 = -1

∴ The correct option is (4).