Trigonometry


  1. If cosx . cosy + sinx . siny = –1 then cosx + cosy is









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    cosx . cosy + sinx. siny = –1
    ⇒ cosx . cosy + 1 = – sinx . siny On squaring both sides, (cosx . cosy + 1)² = sin²x sin²y
    ⇒ cos²x . cos²y + 2cosx . cosy + 1 = (1 – cos²x) (1 – cos²y)
    ⇒ cos²x . cos2y + 2 cosx. cosy + 1 = 1 – cos²x – cos²y + cos²x . cos²y
    ⇒ cos²x + cos2y + 2cosx . cosy = 0
    ⇒ (cosx + cosy)² = 0
    ⇒ cosx + cosy = 0

    Correct Option: C

    cosx . cosy + sinx. siny = –1
    ⇒ cosx . cosy + 1 = – sinx . siny On squaring both sides, (cosx . cosy + 1)² = sin²x sin²y
    ⇒ cos²x . cos²y + 2cosx . cosy + 1 = (1 – cos²x) (1 – cos²y)
    ⇒ cos²x . cos2y + 2 cosx. cosy + 1 = 1 – cos²x – cos²y + cos²x . cos²y
    ⇒ cos²x + cos2y + 2cosx . cosy = 0
    ⇒ (cosx + cosy)² = 0
    ⇒ cosx + cosy = 0


  1. If sin P + cosec P = 2, then the value of sin7P + cosec7P is









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    sinP + cosec P = 2

    ⇒ sinP +
    1
    = 2
    sin P

    sin²P + 1
    = 2
    sin P

    ⇒ sin²P + 1 = 2 sinP
    ⇒ sin²P – 2sin P + 1 = 0
    ⇒ (sin P – 1)2 = 0
    ⇒ sinP – 1 = 0
    ⇒ sinP = 1
    ∴ cosec P = 1
    ∴ sin7P + cosec7P
    = 1 + 1 = 2

    Correct Option: B

    sinP + cosec P = 2

    ⇒ sinP +
    1
    = 2
    sin P

    sin²P + 1
    = 2
    sin P

    ⇒ sin²P + 1 = 2 sinP
    ⇒ sin²P – 2sin P + 1 = 0
    ⇒ (sin P – 1)2 = 0
    ⇒ sinP – 1 = 0
    ⇒ sinP = 1
    ∴ cosec P = 1
    ∴ sin7P + cosec7P
    = 1 + 1 = 2



  1. If secA + tanA = a, then the value of cosA is









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    secA + tanA = a ..... (i)
    ∵ sec²A – tan²A = 1
    ⇒ (secA + tanA) (secA – tanA) = 1
    ² secA – tanA = 1/a ... (ii)
    On adding equations (i) and (ii),

    2 secA = a +
    1
    =
    a² + 1
    aa

    ⇒ secA =
    a² + 1
    2a

    ⇒ cosA =
    2a
    a² + 1

    Correct Option: B

    secA + tanA = a ..... (i)
    ∵ sec²A – tan²A = 1
    ⇒ (secA + tanA) (secA – tanA) = 1
    ² secA – tanA = 1/a ... (ii)
    On adding equations (i) and (ii),

    2 secA = a +
    1
    =
    a² + 1
    aa

    ⇒ secA =
    a² + 1
    2a

    ⇒ cosA =
    2a
    a² + 1


  1. What is the value of sin
    11π
    ?
    6









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    sin
    11π
    6

    = sin 2π -
    π
    6

    [∵ sin (360° – θ)
    = sin (2π – θ) = –sin θ]
    = - sin
    π
    = -
    1
    62

    Correct Option: C

    sin
    11π
    6

    = sin 2π -
    π
    6

    [∵ sin (360° – θ)
    = sin (2π – θ) = –sin θ]
    = - sin
    π
    = -
    1
    62



  1. If
    1
    = x, then the value of x is
    (tanA + cotA)









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    1
    = x
    tanA + cotB

    1
    = x
    sinA
    +
    cosA
    cosAsinA

    1
    = x
    sin²A + cos²A
    sinAcosA

    1
    = sinA.cosA
    1
    sinAcosA

    Correct Option: A

    1
    = x
    tanA + cotB

    1
    = x
    sinA
    +
    cosA
    cosAsinA

    1
    = x
    sin²A + cos²A
    sinAcosA

    1
    = sinA.cosA
    1
    sinAcosA