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    Trigonometric expressions

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    1) 2 tan (theta /2)=

    2) 2 tan theta Sin ^2 (theta /2)=

    3) 2 Cos (45 + x) Cos (45-x)=

    4) (Sin 3 x - 3 Sin x )/ ( Cos 3 x + 3 Cos x) =

    5) 2 tan (theta /2) / { 1 + tan ^2 (theta /2) } =

    6) Find the least positive value of theta such that
    tan (45+ theta ) - 3 tan theta =2

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    https://brainmass.com/math/trigonometry/trigonometric-expressions-simplified-43190

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    Note:
    ^ denotes power and * denotes multiplication

    1) 2 tan (theta /2)=

    From trigonometric identities
    tan (alpha +beta ) = (tan alpha + tan beta ) / ( 1- tan alpha tan beta )
    Putting alpha =beta =theta /2 in the identity
    tan (theta /2+theta /2) = (tan theta /2 + tan theta /2) / ( 1- tan theta /2 tan theta /2)

    or tan theta = 2 tan theta /2 / (1- tan ^2 theta /2)

    or 2 tan theta /2 = tan theta * (1- tan ^2 theta /2)

    Alternatively:
    tan theta /2 = Sin theta /2 / Cos theta /2
    Multiplying both the numerator and denominator of RHS by Cos theta /2
    tan theta /2 = (Sin theta /2 Cos theta /2) / (Cos theta /2 Cos theta /2)
    Multiplying both sides by 2
    2 tan theta /2 = (2 Sin theta /2 Cos theta /2) / (Cos theta /2 Cos theta /2)
    But Sin theta = 2 Sin theta /2 Cos theta /2

    Cos theta = Cos (theta /2 + theta /2 ) = Cos ^2 theta /2 - Sin ^2 theta /2
    or 1 + Cos theta = 1 + Cos ^2 theta /2 - Sin ^2 theta /2
    But 1- Sin^2 theta /2 = Cos ^2 theta /2
    Therefore 1 + Cos theta = 2 Cos ^2 theta /2
    or Cos ^2 theta = (1 + Cos theta )/2

    Substituting Sin theta for 2 Sin theta /2 Cos theta /2
    and (1 + Cos theta )/2 =Cos ^2 theta

    We get
    2 tan (theta /2) = Sin theta / ( 1+ Cos theta ) /2
    or
    2 tan (theta /2) = 2 Sin theta / ( 1+ Cos theta )

    Answer: 2 tan (theta /2) = 2 Sin theta / ( 1+ Cos theta )

    2) 2 tan theta Sin ^2 (theta /2)=

    Cos theta = Cos^2 (theta /2) - Sin ^2 (theta /2) (Identity)
    Subtracting both left hand side (LHS) and right hand side( RHS) from 1
    or 1- Cos theta = 1- {Cos^2 (theta /2) - Sin ^2 (theta ...

    Solution Summary

    Simplifies trigonometric expressions are examined.

    $2.49

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