Find the value of (tan20°)²/(cosec70°)² + (cot20°)²/(sec70°)² + 2tan15° tan45° tan75°.
Show answer & explanation
Correct answer: B
Explanation
Given:
(tan20°)²/(cosec70°)² + (cot20°)²/(sec70°)² + 2tan15°tan45°tan75°
By the formula:
cosec70° = 1/sin70° = 1/cos20°
sec70° = 1/cos70° = 1/sin20°
tan15°tan75° = 1
tan45° = 1
(tan20°)²/(cosec70°)²
= tan²20°/(1/cos²20°)
= tan²20° × cos²20°
= (sin²20°/cos²20°) × cos²20°= sin²20°
Now
(cot20°)²/(sec70°)²
= cot²20°/(1/sin²20°)
= cot²20° × sin²20°
= (cos²20°/sin²20°) × sin²20°= cos²20°
Now
sin²20° + cos²20° = 1
And
2tan15°tan45°tan75°= 2 × 1 × 1= 2
Therefore
1 + 2 = 3