Homework Solution: Simplifying Trig Identity - cot2xsecx + 1/cosx

In summary, the question is asking to simplify the expression cot2xsecx + 1/cosx, and it can be simplified to \csc^2 {x} \sec{x}. This can be easily shown by using the identity \cot^2{x} + 1 = \csc^2 {x}.
  • #1
Kiff
6
0
Help please on trig identity

Homework Statement


Simplify cot2xsecx + 1/cosx





The Attempt at a Solution


Well so far i got:
cot2xsecx + 1/cosx
=(cos2x/sin2x)(1/cosx) + 1/cosx
=((1+cos2x)/(1-cos2x))(1/cosx) + 1/cosx


and from there I am stuck, I've tried playing around with it but I just seem to get to dead ends.
any help is appreciated, Thanks
 
Last edited:
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  • #2


In your second step cancel the square of the cosine. Then merge the entire term into one fraction.
 
  • #3


wow that was fast, thanks

so by cancelling the square I get:
(cosx/sinx)(1/cosx) + 1/cosx
=sinx/cos2x + 1/cosx <--- multiply by cosx
=sinx/cos2x + cosx/cos2x
=sinx + cosx

did i do that correctly?
the question I am doing is multiple choice answer tho
a. csc2xsecx
b. sec3x
c. sec2xcscx
d. csc3x
 
  • #4


[tex] \cot^2{x} \sec{x} + {1 \over \cos{x}} [/tex]

= [tex] \cot^2{x}\sec{x} + \sec{x} [/tex]

= [tex](\cot^2{x} + 1) \sec{x} [/tex]

= [tex]\csc^2 {x} \sec{x} [/tex]

If you are familiar with the identity, [tex] \cot^2{x} + 1 = \csc^2 {x} [/tex], it's obvious.

EDIT: In case you aren't familiar with that particular identity, it's simple to show with a couple more steps:
[tex] 1 + cot^2{u} [/tex]

=[tex]1 + {cos^2{u} \over sin^2{u}} [/tex]

=[tex]{sin^2{u} + cos^2{u} \over sin^2{u}} [/tex]

=[tex]{1 \over \sin^2{u} } [/tex]

=[tex]\csc^2{u} [/tex]
 
Last edited:
  • #5


Also, in you solution, this step is not valid:

=sinx/cos2x + cosx/cos2x
=sinx + cosx
 
  • #6


awsome..thanks guys, omfg i forgot bout that identity
 

What are trigonometric identities?

Trigonometric identities are equations that involve trigonometric functions (such as sine, cosine, and tangent) and are true for all values of the variables in the equation. They are useful for simplifying and solving trigonometric equations.

What is the purpose of using trigonometric identities?

The main purpose of using trigonometric identities is to simplify and solve trigonometric equations. They can also be used to prove other mathematical theorems and to convert between different forms of trigonometric expressions.

How do I know which trigonometric identity to use?

The trigonometric identity to use depends on the specific problem you are trying to solve. It is important to have a good understanding of the available identities and to practice using them to become familiar with their applications.

What are the most commonly used trigonometric identities?

Some of the most commonly used trigonometric identities include the Pythagorean identities, sum and difference identities, double angle identities, and half angle identities. It is also important to be familiar with the reciprocal and quotient identities.

How can I check if my solution using a trigonometric identity is correct?

To check if your solution using a trigonometric identity is correct, you can substitute the values of the variables back into the original equation and see if it satisfies the equation. You can also use a calculator to check if the values are equivalent on both sides of the equation.

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