Trigonometry
- If tan θ + cot θ = 5, then tan²θ + cot²θ is
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tanθ + cotθ = 5
On squaring both sides, (tanθ + cotθ)² = 52
⇒ tan² θ + cot² θ + 2 tanθ.cotθ = 25
⇒ tan² θ + cot² θ = 25 – 2 = 23 [∵ tanθ.cotθ = 1]Correct Option: A
tanθ + cotθ = 5
On squaring both sides, (tanθ + cotθ)² = 52
⇒ tan² θ + cot² θ + 2 tanθ.cotθ = 25
⇒ tan² θ + cot² θ = 25 – 2 = 23 [∵ tanθ.cotθ = 1]
- The value of sin²22° + sin²68° + cot²30° is
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sin²22° + sin²68° + cot²30°
= sin²22° + sin² (90° – 22°) + (√3)²
= sin²22° + cos²22° + 3 [∵ sin²θ + cos²θ = 1]
= 1 + 3 = 4Correct Option: A
sin²22° + sin²68° + cot²30°
= sin²22° + sin² (90° – 22°) + (√3)²
= sin²22° + cos²22° + 3 [∵ sin²θ + cos²θ = 1]
= 1 + 3 = 4
- The minimum value of 2sin²θ + 3cos²θ is
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2 sin²θ + 3 cos²θ
= 2 sin²θ + 2cos²θ + cos²θ
= 2 (sin²θ + cos²θ) + cos²θ
= 2 + cos²θ
∴ Minimum value = 2 + 0 = 2 because cos²θ ≥ 0Correct Option: C
2 sin²θ + 3 cos²θ
= 2 sin²θ + 2cos²θ + cos²θ
= 2 (sin²θ + cos²θ) + cos²θ
= 2 + cos²θ
∴ Minimum value = 2 + 0 = 2 because cos²θ ≥ 0
- If A, B, and C be the angles of a triangle, then out of the following, the incorrect relation is :
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In a triangle ABC,
A + B + C = 180°⇒ A + B + C = 90° 2 2 2 ⇒ A + B = 90° - C 2 2 ⇒ tan 
A + B 
2 = tan 90° - C = cot C 2 2
Correct Option: B
In a triangle ABC,
A + B + C = 180°⇒ A + B + C = 90° 2 2 2 ⇒ A + B = 90° - C 2 2 ⇒ tan A + B 2 = tan 90° - C = cot C 2 2
- If 5 sinθ = 3, the numerical value of
secθ - tanθ is secθ + tanθ
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5 sinθ = 3 ⇒ sinθ = 3 5 Expression = sec θ - tan θ sec θ + tan θ 
= 2 = 1 8 4
Correct Option: D
5 sinθ = 3 ⇒ sinθ = 3 5 Expression = sec θ - tan θ sec θ + tan θ = 2 = 1 8 4
