Trigonometry


  1. What is the value of
    (cotθ + cosecθ - 1)
    ?
    (cotθ - cosecθ + 1)










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    Expression =
    (cotθ + cosecθ - 1)
    (cotθ - cosecθ + 1)

    Expression =
    cotθ + cosecθ - (cosec2θ - cot2θ)
    (cotθ - cosecθ + 1)

    Expression =
    (cotθ + cosecθ) - (cosecθ - cotθ)(cosecθ + cotθ)
    (cotθ - cosecθ + 1)

    Expression =
    (cotθ + cosecθ)(1 - cosecθ + cotθ)
    = cotθ + cosecθ
    (cotθ - cosecθ + 1)

    Correct Option: A

    Expression =
    (cotθ + cosecθ - 1)
    (cotθ - cosecθ + 1)

    Expression =
    cotθ + cosecθ - (cosec2θ - cot2θ)
    (cotθ - cosecθ + 1)

    Expression =
    (cotθ + cosecθ) - (cosecθ - cotθ)(cosecθ + cotθ)
    (cotθ - cosecθ + 1)

    Expression =
    (cotθ + cosecθ)(1 - cosecθ + cotθ)
    = cotθ + cosecθ
    (cotθ - cosecθ + 1)


  1. Find the value of 8 cos 10° cos20° cos 40°.









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    Expression = 8 cos 10°. cos 20° . cos 40°

    Expression = 4
    2 sin 10° .cos 10°. cos 20° . cos 40°
    sin 10°

    Expression = 2
    2 sin 20° . cos 20° . cos 40°
    sin 10°

    { ∴ 2 sinθ . cosθ = sin2θ }
    Expression =
    2 sin 40° . cos 40°
    sin 10°

    Expression =
    sin 80°
    =
    sin 80°
    sin 10°cos (90° - 10°)

    Expression =
    sin 80°
    or
    cos 10°
    cos 80°sin 10°

    Expression = tan80° or cot 10°
    { ∴ cos (90° - θ) = sinθ and sin (90° - θ) = cosθ }

    Correct Option: C

    Expression = 8 cos 10°. cos 20° . cos 40°

    Expression = 4
    2 sin 10° .cos 10°. cos 20° . cos 40°
    sin 10°

    Expression = 2
    2 sin 20° . cos 20° . cos 40°
    sin 10°

    { ∴ 2 sinθ . cosθ = sin2θ }
    Expression =
    2 sin 40° . cos 40°
    sin 10°

    Expression =
    sin 80°
    =
    sin 80°
    sin 10°cos (90° - 10°)

    Expression =
    sin 80°
    or
    cos 10°
    cos 80°sin 10°

    Expression = tan80° or cot 10°
    { ∴ cos (90° - θ) = sinθ and sin (90° - θ) = cosθ }



  1. If secθ + tanθ = 2, then the value of sinθ is :









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    secθ + tanθ = 2
    ∴ sec2θ – tan2θ = 1
    ⇒ (secθ + tanθ)(secθ – tanθ) = 1

    ⇒ secθ – tanθ =
    1
    2

    ∴ secθ + tanθ + secθ – tanθ = 2 +
    1
    2

    ⇒ 2secθ =
    5
    ⇒ secθ =
    5
    24

    Again, (secθ + tanθ) – (secθ – tanθ) = 2 -
    1
    2

    ⇒ 2tanθ =
    3
    ⇒ tanθ =
    3
    24

    ⇒ sinθ =
    tanθ
    =
    3
    ÷
    5
    =
    3
    secθ445

    Correct Option: D

    secθ + tanθ = 2
    ∴ sec2θ – tan2θ = 1
    ⇒ (secθ + tanθ)(secθ – tanθ) = 1

    ⇒ secθ – tanθ =
    1
    2

    ∴ secθ + tanθ + secθ – tanθ = 2 +
    1
    2

    ⇒ 2secθ =
    5
    ⇒ secθ =
    5
    24

    Again, (secθ + tanθ) – (secθ – tanθ) = 2 -
    1
    2

    ⇒ 2tanθ =
    3
    ⇒ tanθ =
    3
    24

    ⇒ sinθ =
    tanθ
    =
    3
    ÷
    5
    =
    3
    secθ445


  1. ∠Y is the right angle of the trianlge XYZ. If XY = 2 √6 cm and XZ – YZ = 2cm, then the value of (secX + tanX) is :









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    XY = 2 6 cm
    XY – YZ = 2 cm. ...(i)
    ∴ XZ2 = XY2 + YZ2
    ⇒ XZ2 - YZ2 = (2√6)2
    ⇒ XZ2 - YZ2 = 24

    XZ2 - YZ2
    =
    24
    XZ - YZ2

    ⇒ XZ + YZ = 12 ....(ii)
    ⇒ secX + tan X =
    XZ
    +
    YZ
    XYXY

    ⇒ secX + tan X =
    XZ + YZ
    =
    XY

    ⇒ secX + tan X =
    XZ + YZ
    =
    12
    = √6
    XY2√6

    Correct Option: D


    XY = 2 6 cm
    XY – YZ = 2 cm. ...(i)
    ∴ XZ2 = XY2 + YZ2
    ⇒ XZ2 - YZ2 = (2√6)2
    ⇒ XZ2 - YZ2 = 24

    XZ2 - YZ2
    =
    24
    XZ - YZ2

    ⇒ XZ + YZ = 12 ....(ii)
    ⇒ secX + tan X =
    XZ
    +
    YZ
    XYXY

    ⇒ secX + tan X =
    XZ + YZ
    =
    XY

    ⇒ secX + tan X =
    XZ + YZ
    =
    12
    = √6
    XY2√6



  1. If α + θ =
    and tanθ = √3 , then the value of tanθ is :
    12










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    tanθ = √3 = tan
    π
    3

    ⇒ θ =
    π
    3

    ∴ α + θ =
    12

    ⇒ α =
    -
    π
    123

    ⇒ α =
    7π - π
    =
    π
    124

    ∴ tanα = tan
    π
    = 1
    4

    Correct Option: B

    tanθ = √3 = tan
    π
    3

    ⇒ θ =
    π
    3

    ∴ α + θ =
    12

    ⇒ α =
    -
    π
    123

    ⇒ α =
    7π - π
    =
    π
    124

    ∴ tanα = tan
    π
    = 1
    4