Question

Find the value of :

 [cosec²(90 - θ ) - tan²θ ] / [3(cosec²67° - tan²23°)] + [3cosec²60° . tan²28° .tan²62] / [3(cos²20 + cos²70)]

1. 2/3
2. 5/3
3. 4/3
4. None of these

Answer 1

Correct Answer - Option 2 : 5/3

Given∶

A trigonometric expression  [cosec2(90 - θ ) - tan2θ ] / [3(cosec267° - tan223°)] + [3cosec260° . tan228° .tan262] / [3(cos220 + cos270)].

Formula Used∶

Trigonometric ratios of some particulars angles,  cosec 60

°= 2/

√3

Trigonometric ratios of complementary angles.

Trigonometric Identities , sec²A - tan² A = 1, sin²A + cos²A = 1

Calculation∶

 

[cosec²(90 - θ) - tan²θ] / [3(cosec²67°- tan²23° )] + [(3cosec²60.tan²28.tan²62)] / [3(cos²20 + cos²70)]

⇒ [sec²θ - tan²θ] / [3{cosec²(90 - 23)} - tan²23}] + 3(2/√3)².tan²28.tan²(90 - 28) / 3[cos²(90 - 70) + cos²70]

⇒  1 / 3[sec²23 - tan²23] + [(3 × 4/3) . tan²28.sec²28] / 3[sin 70 + cos²70]

⇒ [1/3(1)] + [{4.(1)} / 3(1)]

⇒ 1/3 + 4/3 = 5/3

∴ The correct option is (2).