Proving of Identities

Answer the following: cos(θ − β) = tanθ + cotβ is an identity. cosθ sinβ

1. Show that

2. Prove that sin(x + y) + sin(x − y) = 2 sinx cosy.

3. Verify that

sin(x − y) tanx − tany = sin(x + y) tanx + tany

4. Derive an identity for cos3θ in terms of cosθ.

5. Prove that

sin2θ + sinθ = tanθ is an identity. cos2θ + cosθ + 1 2tanθ = sin2θ is an identity. 1 + tan2 θ

6. Verify theta

7. Prove that

x 2 − tan2 = 1 is an identity. 1 + cosx 2

8. Show

tanθ − sinθ θ = sin2 is an identity. 2tanθ 2

θ 2 = cosθ. 9. Prove that θ 1 + tan2 2 1 − tan2 tanθ + sinθ θ = cos2 is an identity. 2tanθ 2

10. verify that

1

1.1

Solutions

cos(θ − β) = tanθ + cotβ is an identity. cosθ sinβ cos(θ − β) cosθ sinβ = = = = cosθcosβ + sinθsinβ cosθsinβ sinθsinβ cosθcosβ + cosθsinβ cosθsinβ sinθ cosβ + sinβ cosθ cotβ + tanθ

1. Show that

2. Prove that sin(x + y) + sin(x − y) = 2 sinx cosy. sin(x + y) + sin(x − y) (sinx cosy + cosx siny) + (sinx cosy − cosx siny) (sinx cosy + sinx cosy) + (cosx siny − cosx siny) 2sinx cosy

= = =

3. Verify that

sin(x − y) tanx − tany = sin(x + y) tanx + tany 1 sinx cosy − cosx siny cosx cosy • 1 sinx cosy + cosx siny cosx cosy sinx cosy cosx siny − cosx cosy cosx cosy cosx siny sinx cosy + cosx cosy cosx cosy sinx siny − cosx cosy sinx siny + cosx cosy tanx − tany tanx + tany

sin(x − y) sin(x + y)

=

=

=

=

4. Derive an identity for cos3θ in terms of cosθ.

cos3θ

= = = = = = = =

cos(2θ + θ) cos2θ cosθ − sin2θ sinθ (1 − 2sin2 θ)cosθ − 2sinθcosθsinθ cosθ − 2sin2 θcosθ − 2sin2 θcosθ cosθ − 4sin2 θcosθ cosθ − 4(1 − sin2 θ)cosθ cosθ − 4cosθ + 4cos3 θ − 3cosθ + 4cos3 θ

2

5. Prove that

sin2θ + sinθ = tanθ is an identity. cos2θ + cosθ + 1 sin2θ + sinθ cos2θ + cosθ + 1 = = = = = 2sinθcosθ + sinθ 2cos2 θ − 1 + cosθ + 1 2sinθcosθ + sinθ 2cos2 θ + cosθ sinθ(2cosθ + 1) cosθ(2cosθ + 1) sinθ cosθ tanθ

6. Verify that

2tanθ = sin2θ is an identity. 1 + tan2 θ 2tanθ 1 + tan2 θ 2sinθ cosθ sec2 θ 2sinθ 1 • cosθ sec2 θ 2sinθ • cos2 θ cosθ 2sinθ • cosθ 2sinθ cosθ

= = = = =

7. Prove that

2 x − tan2 = 1 is an identity. 1 + cosx 2

2 x − tan2 1 + cosx 2

= = = = = =

2 − ( 1 + cosx

1 − cosx 2 ) 1 + cosx

2 1 − cosx − 1 + cosx 1 + cosx 2 − (1 − cosx) 1 + cosx 2 − 1 + cosx 1 + cosx 1 + cosx 1 + cosx 1

3

8. Show

tanθ − sinθ θ = sin2 is an identity. 2tanθ 2

tanθ − sinθ 2tanθ

=

= = = = θ 2 = cosθ. 9. Prove that θ 1 + tan2 2 1 − tan2

sinθ − sinθ cosθ 2sinθ cosθ sinθ − sinθcosθ cosθ • cosθ 2sinθ sinθ(1 − cosθ) cosθ • cosθ 2sinθ 1 − cosθ 2 θ sin2 2

θ 2 θ 1 + tan2 2 1 − tan2

=

1 − cosx 1 + cosx 1 − cosx 1 + 1 + cosx 1 − 1 + cosθ − 1 + cosθ 1 + cosθ 1 + cosθ + 1 − cosθ 1 + cosθ 2cosθ 1 + cosθ • 1 + cosθ 2 cosθ

=

= = 10. Verify that tanθ + sinθ θ = cos2 is an identity. 2tanθ 2

tanθ + sinθ 2tanθ

=

= = = =

sinθ + sinθ cosθ 2sinθ cosθ sinθ + sinθcosθ cosθ • cosθ 2sinθ sinθ(1 + cosθ) cosθ • cosθ 2sinθ 1 + cosθ 2 cos2 θ 2

4

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