Cumulative Assessment
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Review the trigonometric identities.
| Expression | Equivalent |
|---|---|
| tan x sec x cos x | No |
| sin^2x+cos^2x | Yes |
| cos^2(- x) tan^2 x/sin^2(- x) | Yes |
| cos (π/2-x) csc x | Yes |
We are asked to find which of the given expressions are equivalent to 1. For that, we are going to analyze and simplify each of the expressions separately.
sec x= 1/cos x
a/cos x* cos x = a
Identity Property of Multiplication
We found that, when simplified, the expression equals tan x, which is not always equal to 1. Let's take a look at some examples.
| x | tan x |
|---|---|
| π/6 | sqrt(3)/3 |
| π/4 | 1 |
| π/3 | sqrt(3) |
This means that the expression is not equivalent to 1.
Let's take a look at the expression. sin^2x+cos^2x To see if it is equivalent to 1, we can recall one of the Pythagorean identities. sin^2 θ+cos^2θ=1 The identity tells us that the expression sin^2 θ+cos^2θ is constant and equal to 1, regardless of the value of θ. In our expression we have θ=x. Since the argument does not matter, the expression sin^2x+cos^2x is also equivalent to 1.
We will now move on to the third expression. We want to know the value of the given expression when simplified. cos^2(- x) tan^2 x/sin^2(- x) We will use the three following identities to simplify the expression.
| Negative Angle Identity | Tangent Identity |
|---|---|
| sin (- θ)=- sin θ | tan θ = sin θ/cos θ |
| cos (- θ) = cos θ |
cos (- x)= cos x
sin (- x)= - sin x
(- a)^2=a^2
a* b/c=a/c* b
tan x= sin x/cos x
(a/b)^m=a^m/b^m
Cancel out common factors
Simplify quotient
Let's take a look at the last expression. cos (π/2-x) csc x In order to simplify it, we will use the following cofunction and reciprocal identities.
| Cofunction Identity | Reciprocal Identity |
|---|---|
| cos (π/2-θ )=sin θ | csc θ =1/sin θ |
cos (π/2-x)= sin x
csc x= 1/sin x
sin x * a/sin x= a