# Verifying Identity: Sec(x)Sin2(x) = 1 - cos(x)

• mcca408
In summary, to verify the identity sin2(x)/cos(x)/(1+1/cos(x)) = 1 - cos(x), you can convert all terms to sine and cosine functions and then simplify to get sin2(x)/cos(x) = sin2(x)/cos(x).
mcca408

## Homework Statement

verify the following identity:

Sec(x)Sin2(x)
______________________ = 1 - cos(x)

1 + sec(x)

## Homework Equations

sec(x)=1/cos(x)
sin2(x)=1-cos2(x)

## The Attempt at a Solution

I never know how to start off these problems. I have to take the left side and show that it equals the right by doing trig identities. I try several steps and keep going through an endless loop. I believe I must start of by multiplying 1-sec(x)/1-sec(x)
That way i get

(1-sec(x))(sec(x)sin2(x))
__________________________________

1 - sec2(x)

I'm not sure if I'm starting off correctly

It would probably be easier to start by replacing the sec(x) terms on the left side with 1/cos(x).

Sec(x)Sin2(x)
______________________ = 1 - cos(x)

1 + sec(x)

ok using the identity sec(x)=1/cos(x)

I get

1/cos(x) * sin2(x)
__________________________

1 + 1/cos(x)

and that =

sin2(x)/cos(x)
______________________

1 + 1/cos(x)

multiply top and bottom by cos(x)

Sin2(x) / 2

Did i do anything wrong?

Thanks

mcca408 said:
Did i do anything wrong?
Yes. cos(x)*(1 + 1/cos(x)) is not equal to 2. Try again.

I think a trick in handling trigo qns is to convert all to sine and cosine functions.
since sec, csc and even tan functions can be derived from sine and cosine.

Do remember this to help in your future sch works as well.

#### Attachments

• partial sol.jpg
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mcca408 said:
sin2(x)/cos(x)
______________________

1 + 1/cos(x)

Turn the denominator into $$\frac{cos(x)+1}{cos(x)}$$.

So that would look like $$\frac{\frac{sin^{2}(x)}{cos(x)}}{\frac{cos(x)+1}{cos(x)}}.$$

What icystrike showed is the faster way but since you're already this far, you can try what I suggested.

## 1. What is the identity being verified in the equation "Sec(x)Sin2(x) = 1 - cos(x)"?

The identity being verified is the trigonometric identity for the secant and sine functions, which states that secant multiplied by twice the sine of an angle is equal to one minus the cosine of that same angle.

## 2. How is this identity verified?

This identity can be verified by substituting any value for x into both sides of the equation and confirming that they are equal. It can also be verified algebraically by using the reciprocal and double angle identities for the secant and sine functions.

## 3. What does this identity tell us about the relationship between the secant and sine functions?

This identity tells us that the secant and sine functions are related in such a way that one can be expressed in terms of the other. It also shows that the two functions are inversely proportional, meaning that as one increases, the other decreases.

## 4. Why is verifying identities important in mathematics?

Verifying identities is important because it allows us to confirm the validity of mathematical statements and equations. It also helps us to understand the relationships between different mathematical concepts and can be used to simplify complex expressions.

## 5. Can this identity be applied in real-world situations?

Yes, this identity can be applied in various real-world situations, such as in physics and engineering, where trigonometric functions are used to model and solve problems. It can also be used in navigation, astronomy, and other fields where angles and distances are involved.

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