Trigonometry
- If θ is a positive acute angle and cosec θ = √3 , then the value of cot θ – cosec θ is
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cosecθ = √3
cotθ = √cosec² θ - 1
=(√√3² - 1) = √3 - 1 = √2
∴ cotθ - cosecθ = √2 - √3= 3√2 - √2 = (√2 - √3) 3
Correct Option: E
cosecθ = √3
cotθ = √cosec² θ - 1
=(√√3² - 1) = √3 - 1 = √2
∴ cotθ - cosecθ = √2 - √3= 3√2 - √2 = (√2 - √3) 3
- If q is a positive acute angle and 4 cos²θ – 4 cos θ + 1 = 0, then the value of tan (θ – 15° ) is equal to
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4cos²θ – 4cosθ + 1 = 0
⇒ (2 cosθ – 1)² = 0
⇒ 2 cosθ – 1 = 0
⇒ 2 cosθ = 1⇒ cosθ = 1 = cos 60° 2
⇒ θ = 60°
∴ tan (θ –15°) = tan (60° –15°) =
tan 45° = 1Correct Option: B
4cos²θ – 4cosθ + 1 = 0
⇒ (2 cosθ – 1)² = 0
⇒ 2 cosθ – 1 = 0
⇒ 2 cosθ = 1⇒ cosθ = 1 = cos 60° 2
⇒ θ = 60°
∴ tan (θ –15°) = tan (60° –15°) =
tan 45° = 1
- If (r cos θ – √3 )² + (r sin θ –1)² = 0 then the value of
r tan θ + sec θ is equal to r sec θ + tan θ
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(r cos θ – √3 )⇒ + (r sin θ –1)⇒ = 0
⇒ r cos θ – √3 = 0 and r sin θ – 1 = 0
⇒ r cos θ = √3 and r sin θ = 1
∴ r² cos²θ + r² sin²θ = 3 + 1
⇒ r² (sin²θ + cos²θ) = 4
⇒ r² = 4
⇒ r = 2∴ tan θ = r sin θ = 1 r cos θ √3 and r cosθ = √3 ⇒ cosθ = √3 r ⇒ sec θ = r √3 
= 2r = 2 × 2 = 4 r² + 1 4 + 1 5
Correct Option: A
(r cos θ – √3 )⇒ + (r sin θ –1)⇒ = 0
⇒ r cos θ – √3 = 0 and r sin θ – 1 = 0
⇒ r cos θ = √3 and r sin θ = 1
∴ r² cos²θ + r² sin²θ = 3 + 1
⇒ r² (sin²θ + cos²θ) = 4
⇒ r² = 4
⇒ r = 2∴ tan θ = r sin θ = 1 r cos θ √3 and r cosθ = √3 ⇒ cosθ = √3 r ⇒ sec θ = r √3 = 2r = 2 × 2 = 4 r² + 1 4 + 1 5
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The value of sin 25° cos 65° + cos 25° sin 65° tan²70° - cosec²20°
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sin25°cos65° + cos25°sin65° tan² 70° - cosec² 20°
sin25°cos(90° - 25°) + cos25°sin(90° - sin25°) tan² 70° - cosec² (90° - 70°) 
sin25°sin25° + cos25°.cos25° tan² 70° - sec² 70° sin²25° + cos²25° tan² 70° - sec² 70° = 1 = - 1 - 1 
Correct Option: A
sin25°cos65° + cos25°sin65° tan² 70° - cosec² 20°
sin25°cos(90° - 25°) + cos25°sin(90° - sin25°) tan² 70° - cosec² (90° - 70°) sin25°sin25° + cos25°.cos25° tan² 70° - sec² 70° sin²25° + cos²25° tan² 70° - sec² 70° = 1 = - 1 - 1
- If sin (θ + 18°) = cos 60° (0 < θ < 90°), then the value of cos 5θ is
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sin (θ + 18°) = cos 60°
= cos (90° – 30°) = sin 30°
⇒ θ + 18° = 30°
⇒ θ = 30° – 18° = 12°∴ cos5θ = cos 60° = 1 2
Correct Option: A
sin (θ + 18°) = cos 60°
= cos (90° – 30°) = sin 30°
⇒ θ + 18° = 30°
⇒ θ = 30° – 18° = 12°∴ cos5θ = cos 60° = 1 2
