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Trigonometric expressions

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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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

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