Mid-Chapter Quiz
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Start on the left-hand side of the equation and use the Pythagorean, Reciprocal, and Quotient Identities to arrive at the right-hand side.
See solution.
We want to verify the given trigonometric identity. sin θ * sec θ/sec θ - 1=(sec θ + 1) cot θ We will start on the left-hand side and use the Pythagorean, Reciprocal, and Quotient Identities to arrive at the right-hand side. Let's begin by performing some basic operations on the left-hand side.
a/b=a * (sec θ + 1)/b * (sec θ + 1)
(a+b)(a-b)=a^2-b^2
1^a=1
Now let's recall one of the Pythagorean Identities.
sec ^2 θ= tan ^2 θ + 1
1-1=0
Next, we will recall one of the Reciprocal Identities. sec θ = 1/cos θ, cos θ ≠0 We can substitute 1cos θ for sec θ and simplify our expression.
sec θ= 1/cos θ
a* 1/b= a/b
Now let's recall one of the Quotient Identities. tan θ = sin θ/cos θ, cos θ ≠0 We can substitute tan θ for sin θcos θ and simplify our expression.
sin θ/cos θ= tan θ
Finally, we will recall once again one of the Reciprocal Identities. tan θ = 1/cot θ, cot θ ≠0 We can substitute 1cot θ for tan θ and simplify our expression.
We started on the left-hand side of the identity and used the Pythagorean, Reciprocal, and Quotient Identities to arrive at the right-hand side. Therefore, we have verified the identity.