Activity 8: Trigonometric Identities

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MHF4U_2011:Advanced Functions, Grade 12, University Preparation

Unit 4: Trigonometric Functions

Homework/Formative Assessment

1. Simplify the following:

a) 1₂ 1₂ sin xtan x b) sin cos₂

1 sin

x x

 x

2. Prove the following identities. Explain steps:

a) 1 1

tanx tan sin cos

x x x

 

b) 𝑠𝑒𝑐² 𝑥 2 𝑠𝑒𝑐 𝑥 𝑐𝑜𝑠 𝑥 𝑐𝑜𝑠² 𝑥 𝑡𝑎𝑛² 𝑥 𝑠𝑖𝑛² 𝑥

c) sin( )

1 cot tan

sin cos x y

x y

  

d) 1 sin 2 cos 2 cos 2 1 sin 2

x x

 



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Homework/Formative Assessment SOLUTIONS 1. Simplify the following:

a)

2 2

2

2 2

cos sin

1 1

sin tan 1 sin 1 cos

si s n

in sin 1

x x

x x x

x



 



 b)

2

sin cos 1 sin

sin cos si

cos n cos tan

x x

x

x x

x x x x





2. Prove the following identities. Explain steps:

a) KEEP IN MIND THAT THESE STEPS ARE NOT THE ONLY WAY TO PROVE THESE.

2 2

1 1

tan tan sin cos

change to sine and cosine

sin 1

cos sin sin cos

common denominator 1

sin cos

pythagorean identity

1 1

sin co

. . . .

c

s sin os

sin cos sin c

cos

. . .

os

.

x x x x

x

x x x x

x x

L S

x x x x

L S R S

x

S

x x

x

R x

 



 

 



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b)

2

2 2

2 2 2

2

2 2

. . . .

sec ² 2sec cos cos ² tan ² sin ²

change everything to sine and cosine

1 1 sin

2 cos cos sin

cos cos cos

LS simplify

RS common denominator

1 sin sin cos

2 cos

cos cos

LS com L S R S

x x x x x

x x x x

x x x



   

 

     

   

 

  

2 2

2 4

2 2

2 4

2 2

2 4 2 4

2 2

mon denominator RS factor

sin 1 cos 1 2cos cos

cos cos

pythagorean identity 1 cos 1 cos 1 2cos cos

cos cos

expand

1 2cos cos 1 2cos cos

cos cos

. . . .

x x

x x x x

x x

L S R S

   

 

  

   



 

c)

sin( ) 1 cot tan

sin cos

change to sine and cosine use the compound identity cos sin sin cos sin cos

1 sin cos sin cos

RS distribute denominator cos sin sin c

. . . .

1 os

sin cos sin co x y

x y

x y x y y x

x y x y

x y x

L S R S

  

  

 

  

 



sin cos s sin cos simplify

cos sin cos sin

1 1

sin cos sin cos . . . .

y x

y x y

x y x y

L S R S



  

 

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d)

2 2

2 2 2 2

2 2

1 2sin cos cos sin cos sin 1 2sin cos

change 1 into sine and cosi . . . .

1 sin 2 cos 2 cos 2 1 sin 2

ne using pythagorean co

use double

s sin

angl

2sin cos cos sin

co

e identit

s sin cos

ies L S R S

x

x x x x

x x x x x x

x

x x

x

  





 



  

 

  

     

  

2 2

2 2 2 2

sin 2sin cos rearrange for factoring

cos 2sin cos sin cos sin

cos sin cos 2sin cos sin

factor

cos sin cos sin cos sin cos sin

simplify cos si

x x x x

x x x x x x

x x x x x x x x

x

 

   

  

   

    







   

 

n cos sin

cos sin cos sin

. . . .

x x x

x x x x

L S R S

 

 

 