Trigonometric Identities and proofs

Additional maths

Adrian Sparrow

www.ibmaths.com

Using the unit circle

r

€

cos θ = xr

sinθ = yr

tanθ = yx

€

sinθcosθ

=

yr

xr

€

sinθcosθ

= yx

€

yx

= tanθ

€

tanθ= sinθcosθ

Draw the graph of

€

y =sin2 θ

The graph of

€

y =cos2 θ

On the same axis:

Add the graphs together to make the blue line shown below.

€

sin2θ+cos2θ=1

Two more identities starting with

€

sin2θ+cos2θ=1

Divide by Divide by

€

cos2 θ

€

sin2 θ

€

sin2 θcos2 θ

+cos2 θcos2 θ

= 1

cos2 θ

€

sin2 θsin2 θ

+cos2 θsin2 θ

= 1

sin2 θ

€

sinθcosθ

2

+1 = 1cosθ

2

€

1 + cosθsinθ

2

= 1sinθ

2

€

1+tan2θ=sec2θ

€

1+cot2θ=cos ec2θ

Prove the following:

€

1−2sin2 x ≡2cos2 x −1

€

1−2sin2 x ≡1−2 1−cos2 x( )

€

sin2 x +cos2 x =1use the identity

€

1−2sin2 x ≡1−2+2cos2 x

€

1−2sin2 x ≡2cos2 x −1

€

1−tan2 x

1+tan2 x≡cos2 x −sin2 x

€

1 − tan2 x

1 + tan2 x≡

1 − sin2 xcos2 x

1 + sin2 xcos2 x

€

1 −tan2 x

1 + tan2 x≡

cos2 xcos2 x

−

sin2 xcos2 x

cos2 xcos2 x

+

sin2 xcos2 x

€

1−tan2 x

1+tan2 x≡cos2 x −sin2 x

1

€

1−tan2 θ1+tan2 θ

≡cos2 θ−sin2 θcos2 θ+sin2 θ

Try these with no hints

€

sinxsec x −1

+ sinxsec x +1

≡2cot x

€

sinxsec x −1

+ sinxsec x +1

≡2

sinxcos x

tan2 x

€

sinxsec x −1

+ sinxsec x +1

≡2tanx

tan2 x

€

sinxsec x −1

+ sinxsec x +1

≡ 2tanx

€

sinxsec x −1

+ sinxsec x +1

≡2cot x

€

sinxsec x −1

+ sinxsec x +1

=sinx sec x +1( )+sinx sec x −1( )

sec2 x −1( )

€

sinxsec x −1

+ sinxsec x +1

≡sinx sec x +sinx +sinx sec x −sinx

tan2 x