Exercise 1 Page 927

We want to verify the given trigonometric identity. sin θ tan θ = sec θ - cos θ We will start on the left-hand side and use Trigonometric Identities to arrive at the right-hand side. At first, let's recall the Tangent Identity. tan θ = sin θ/cos θWith this identity, we can substitute sin θcos θ for tan θ in our expression.

sin θ tan θ

Substitute

tan θ= sin θ/cos θ

sin θ * sin θ/cos θ

MoveLeftFacToNum

a*b/c= a* b/c

sin^2 θ/cos θ

Now we will use one of the Pythagorean Identities. Note that we will need to rearrange the terms so that we can use it for our expression. sin^2 θ + cos^2 θ = 1 ⇕ sin^2 θ = 1 - cos^2 θ We can substitute 1 - cos^2 θ for sin^2 θ in our expression.

sin^2 θ/cos θ

Substitute

sin^2 θ= 1 - cos^2 θ

1 - cos^2 θ/cos θ

WriteDiffFrac

Write as a difference of fractions

1/cos θ - cos^2 θ/cos θ

ReduceFrac

a/b=.a /cos θ./.b /cos θ.

1/cos θ - cos θ/1

DivByOne

a/1=a

1/cos θ - cos θ

Finally, let's apply one of the Reciprocal Identities. sec θ = 1/cos θ With this identity, we can substitute sec θ for 1cos θ in our expression.

1/cos θ - cos θ

Substitute

1/cos θ= sec θ

sec θ - cos θ ✓

We started on the left-hand side of the identity and used Trigonometric Identities to arrive at the right-hand side. Therefore, we have verified the identity.