Establish the following identities

Question

Establish the following identities. Perform simplifications on only one side and multiplying by convienient forms of 1" to get the other side. Write basic identity used on each appropriate step. You should not perform steps of algebra that change both sides of the identity

 

a) tan(θ)cot(θ)-cos^2(θ)=sin^2(θ)

 

b) 1-sin(θ)/1+sin(θ)=(secθ-tanθ)^2

 

c) sin(θ)+cos(θ)/sin(θ)-cos(θ)-sin(θ)/cos(θ)=sec(θ)csc(θ)

 

Help!

Answer 1

Hi Logan,

 

a. tan(θ)*cot(θ) - cos²θ = ( sin(θ)/cos(θ) ) * ( cos(θ)/sin(θ) ) - cos²θ     (definition of tanθ and cotθ)

 

                                   = 1 - cos²θ                                                      (tanθ*cotθ = 1)

 

                                   = sin²θ                                                            (Pythagorean Identity)

 

b. ( 1 - sin(θ) )/( 1 + sin(θ) )

 

= ( 1 - sin(θ) )/( 1 + sin(θ) )*( ( 1 - sin(θ))/(1 - sin(θ)) )             (multiply by conjugate)

 

= ( 1 - 2sinθ + sin²(θ) )/( 1 - sin²(θ) )                                        (multiply through)

 

= ( 1 / (1 - sin²θ) - 2sinθ/(1 - sin²(θ)) + sin²(θ)/( 1 - sin²(θ) ) )   (seperate fractions)

 

= ( 1/cos²(θ) - 2sinθ/cos²(θ) + sin²(θ)/cos²(θ) )                          (Pythagorean Identity)

 

= sec²(θ) - 2sec(θ) + tan²(θ)                                                      (1/cosθ = secθ)

 

= (sec(θ) - tan(θ))²

 

c. (sin(θ) + cos(θ))/sin(θ) - (cos(θ) - sin(θ))/cos(θ)

 

Get a common denominator sin(θ)cos(θ):

 

= ( (sin(θ) + cos(θ))/sin(θ) )*(cos(θ)/cos(θ)) - ( (cos(θ) - sin(θ))/cos(θ) )*(sin(θ)/sin(θ))

 

= ( sin(θ)cos(θ) + cos²(θ) )/(sin(θ)cos(θ)) - ( cos(θ)sin(θ) - sin²(θ) )/(sin(θ)cos(θ))    (multiply through)

 

= ( sin(θ)cos(θ) + cos²(θ) - cos(θ)sin(θ) + sin²(θ) ) / (sin(θ)cos(θ))                           (add the fractions)

 

= ( cos²(θ) + sin²(θ) )/(sin(θ)cos(θ))                                                                       (cancel terms)

 

= 1/(sin(θ)cos(θ))                                                                                                   (Pythagorean Identity)

 

= csc(θ)sec(θ)                                                                                                        (def of cscθ and secθ)