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# Use trigonometric identities to transform one side of the equation into the other

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1. (1+cosθ)(1-cosθ)=sin^2 θ

2. (sinθ/cosθ)+(cosθ/sinθ)=cscθ secθ

DarkKnight  Jul 1, 2017

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1 -

Verify the following identity:
(cos(θ) + 1) (1 - cos(θ)) = sin(θ)^2

(1 - cos(θ)) (cos(θ) + 1) = 1 - cos(θ)^2:
1 - cos(θ)^2 = ^?sin(θ)^2

sin(θ)^2 = 1 - cos(θ)^2:
1 - cos(θ)^2 = ^?1 - cos(θ)^2

The left hand side and right hand side are identical:
Answer: | (identity has been verified)

2-

Verify the following identity:
(sin(θ))/(cos(θ)) + (cos(θ))/(sin(θ)) = csc(θ) sec(θ)

Put (cos(θ))/(sin(θ)) + (sin(θ))/(cos(θ)) over the common denominator sin(θ) cos(θ): (cos(θ))/(sin(θ)) + (sin(θ))/(cos(θ)) = (cos(θ)^2 + sin(θ)^2)/(cos(θ) sin(θ)):
(cos(θ)^2 + sin(θ)^2)/(cos(θ) sin(θ)) = ^?csc(θ) sec(θ)

Multiply both sides by sin(θ) cos(θ):
cos(θ)^2 + sin(θ)^2 = ^?cos(θ) csc(θ) sec(θ) sin(θ)

Write cosecant as 1/sine and secant as 1/cosine:
cos(θ)^2 + sin(θ)^2 = ^?1/(cos(θ)) 1/(sin(θ)) cos(θ) sin(θ)

cos(θ) (1/(sin(θ))) (1/(cos(θ))) sin(θ) = 1:
cos(θ)^2 + sin(θ)^2 = ^?1

Substitute cos(θ)^2 + sin(θ)^2 = 1:
1 = ^?1

The left hand side and right hand side are identical:
Answer: | (identity has been verified)

Guest Jul 1, 2017
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#1
+2

1 -

Verify the following identity:
(cos(θ) + 1) (1 - cos(θ)) = sin(θ)^2

(1 - cos(θ)) (cos(θ) + 1) = 1 - cos(θ)^2:
1 - cos(θ)^2 = ^?sin(θ)^2

sin(θ)^2 = 1 - cos(θ)^2:
1 - cos(θ)^2 = ^?1 - cos(θ)^2

The left hand side and right hand side are identical:
Answer: | (identity has been verified)

2-

Verify the following identity:
(sin(θ))/(cos(θ)) + (cos(θ))/(sin(θ)) = csc(θ) sec(θ)

Put (cos(θ))/(sin(θ)) + (sin(θ))/(cos(θ)) over the common denominator sin(θ) cos(θ): (cos(θ))/(sin(θ)) + (sin(θ))/(cos(θ)) = (cos(θ)^2 + sin(θ)^2)/(cos(θ) sin(θ)):
(cos(θ)^2 + sin(θ)^2)/(cos(θ) sin(θ)) = ^?csc(θ) sec(θ)

Multiply both sides by sin(θ) cos(θ):
cos(θ)^2 + sin(θ)^2 = ^?cos(θ) csc(θ) sec(θ) sin(θ)

Write cosecant as 1/sine and secant as 1/cosine:
cos(θ)^2 + sin(θ)^2 = ^?1/(cos(θ)) 1/(sin(θ)) cos(θ) sin(θ)

cos(θ) (1/(sin(θ))) (1/(cos(θ))) sin(θ) = 1:
cos(θ)^2 + sin(θ)^2 = ^?1

Substitute cos(θ)^2 + sin(θ)^2 = 1:
1 = ^?1

The left hand side and right hand side are identical:
Answer: | (identity has been verified)

Guest Jul 1, 2017

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