Cambridge International AS and A Level Mathematics Pure Mathematics 1

P1^

7

Exercise

(^) 7B
225
Method 2
When one side of the identity looks more complicated than the other side, you
can work with this side until you end up with the same as the simpler side.
ExAmPlE 7.4 Prove the identity 1 cos−sincθθθ−≡os^1 tanθ.
SOlUTION
The LHS of this identity is more complicated, so manipulate the LHS until you
end up with tan θ.
Write the LHS as a single fraction:
cos
sincos
cos( sin)
cos( sin)
θ
θθ
θθ
1 θθ

1 1

1

2

− −≡

−−

−

≡cosscos(+−−insin)

(^21)
1
θθ
θθ
≡ −+ −
−
1 1
1
sins^2 in
cos( sin)
θθ
θθ
≡ −
−

≡ −

−

sinsin

cos( sin)

sin( sin)

cos( sin

θθ

θθ

θθ

θθ

2

1

1

1 ))

sin

cos

tan

≡

≡

θ

θ

θasrequired

ExERCISE 7B Prove each of the following identities.

1  1 – cos^2 θ ≡ sin^2 θ

2  (1 – sin^2 θ)tan θ ≡ cos θ sin θ

3  12 1

2

sin^2

cos

θ sin

θ

θ

−≡

4  tan

cos

2

2

θ^11

θ

≡−

5  sincos

sincos

22

22

θθ (^312)
θθ

−+

−

≡

6  1122 221

cossθθin cossθθin

+≡

7  tanθ+≡cossinsθθθ in^1 cosθ

(^8 1) +^1 sins+ 1 −^1 in ≡^22
θθ cosθ
9  Prove the identity^1
1
2
2
−

• tan
  tan
  x
  x
  ≡ 1 – 2 sin^2 x.
  [Cambridge AS & A Level Mathematics 9709, Paper 1 Q3 June 2007]
  Since sin^2 θ + cos^2 θ ≡ 1,
  cos^2 θ ≡ 1 – sin^2 θ