Please show ALL WORK in your answer.Thanks

Question

Prove each identity.

39)   sin (-θ) cot(-θ) = cos θ

43)   csc (θ) + sin (-θ) =cos² θ  sin θ

Answer 1

sin(–θ)cot(–θ)=cosθ

L .H.S sin(–θ)cot(–θ)

Even-Odd Identities:sin(–θ)= –sinθ and cot(–θ)= –cotθ

(–sinθ) (–cotθ)

Quotient Identities: cotθ=cosθ/sinθ

(–sinθ)(–cosθ/sinθ)

Cancel common terms.

(–)(–cosθ)

Product of two negative signs are positive.

cosθ=R.H.S

Therefore sin(–θ)cot(–θ)=cosθ.

Answer 2

 

L .H.S sin(–θ)cot(–θ)

Even-Odd Identities:sin(–θ)= –sinθ and cot(–θ)= –cotθ

(–sinθ) (–cotθ)    (Quotient Identities: cotθ=cosθ/sinθ)

(–sinθ)(–cosθ/sinθ)

Cancel common terms.

(–)(–cosθ)

Product of two negative signs are positive.

cosθ=R.H.S

Therefore sin(–θ)cot(–θ)=cosθ.

 

Answer 3

43).

Consider that, the trigonometric equation as csc (θ) + sin (- θ) = cos² (θ) / sin (θ).

Left hand side identity : csc (θ) + sin (- θ).

Reciprocal identity : csc (θ) = 1/sin (θ).

csc (θ) + sin (- θ) = (1/ sin (θ)) + sin (- θ)

Even - Odd Identities : sin(– θ) = – sin (θ).

= (1/ sin (θ)) - sin (θ)

= (1 - sin² θ) / sin (θ)

Pythagoras identity : sin² θ + cos² θ = 1.

= (cos² θ) / sin (θ)

= Right hand side identity.