prove these identities

[icelated]

I have two problems where i am trying to prov there identities.

I think i am close but just not getting it. This isnt really homework. Its just practice problems that are not in the solutions manual.

1. prove the follwoing identities.

b) sec(x)(tan(x) + cot(x)) = csc(x) / cos^2(x)

This is what i have so far,

 = 1/ cos(x) times sin(x) / cos(x) + 1/tan

 = 1/ cos(x) times sin(x) / cos(x) + cos(x) / sin(x)

= 1/ cos(x) times sin^2(x) / cos(x) + cos^2(x) / sin(x)   // just like adding fractions

Since sin^2x + cos^2x = 1

1/cos(x) times 1 / cos(x) sin(x)

1/cos^2(x) sin(x)

this is close to the identity. can i split it up so its 1/cos^2(x) times 1/ sin(x) ?

and now for part c - Note x = theta. i just didnt want to type it out..

c) Tan(x) - cot(x) / tan(x) + cot(x) = sin^2(x) - cos^2(x)

This is what i have so far,

 = sin (x) / cos(x) - cos(x) / sin(x) big divided by cos(x) / cos(x)  + cos(x) / sin(x)

= sin^2(x) - cos^2(x) / cos(x)sin(x) big divided by sin^2(x) + cos^2(x) / cos(x)sin(x)

Now, since cos^2(x) + sin^2(x) = 1

 = 1 / cos(x) - sin(x)  / 1 / cos(x) + sin(x)

=  1 / cos(x) - sin(x)  times  cos(x) + sin(x) / 1

ones cancel out. Now, since theres a negative and a positive between them i cant cancel?
So, i am left with cos(x) + sin(x)  / cos(x) - sin(x)

if i can combine them somehow i have the right identity
What did i do so wrong?

I dont know what to do from here..

 

[Deleted member 4993]

I have two problems where i am trying to prov there identities.

I think i am close but just not getting it. This isnt really homework. Its just practice problems that are not in the solutions manual.

1. prove the follwoing identities.

b) sec(x)(tan(x) + cot(x)) = csc(x) / cos^2(x)

This is what i have so far,

 = 1/ cos(x) times sin(x) / cos(x) + 1/tan

 = 1/ cos(x) times sin(x) / cos(x) + cos(x) / sin(x)

= 1/ cos(x) times sin^2(x) / cos(x) + cos^2(x) / sin(x)   // just like adding fractions

Since sin^2x + cos^2x = 1

1/cos(x) times 1 / cos(x) sin(x)

1/[cos^2(x) sin(x)]

you are almost done...

= 1/cos[sup:28immbxn]2[/sup:28immbxn](x) * 1/sin(x)

= 1/cos[sup:28immbxn]2[/sup:28immbxn](x) * csc(x)

= csc(x)/cos[sup:28immbxn]2[/sup:28immbxn](x)

this is close to the identity. can i split it up so its 1/cos^2(x) times 1/ sin(x) ?

and now for part c - Note x = theta. i just didnt want to type it out..

c) Tan(x) - cot(x) / tan(x) + cot(x) = sin^2(x) - cos^2(x)

This is what i have so far,

 = sin (x) / cos(x) - cos(x) / sin(x) big divided by cos(x) / cos(x)  + cos(x) / sin(x)

= sin^2(x) - cos^2(x) / cos(x)sin(x) big divided by sin^2(x) + cos^2(x) / cos(x)sin(x)

Now, since cos^2(x) + sin^2(x) = 1

 = 1 / cos(x) - sin(x)  / 1 / cos(x) + sin(x)

=  1 / cos(x) - sin(x)  times  cos(x) + sin(x) / 1

ones cancel out. Now, since theres a negative and a positive between them i cant cancel?
So, i am left with cos(x) + sin(x)  / cos(x) - sin(x)

if i can combine them somehow i have the right identity
What did i do so wrong?

I dont know what to do from here..

You are almost finished with this one too - just do your "algebra" cleanly .....

\(\displaystyle \frac{tan(x) - cot(x)}{tan(x)+cot(x)}\)

\(\displaystyle = \ \frac{\frac{sin(x)}{cos(x)} -\frac{cos(x)}{sin(x)}}{\frac{sin(x)}{cos(x)}+\frac{cos(x)}{sin(x)}}\)

\(\displaystyle = \ \frac{\frac{sin^2(x) - cos^2(x)}{cos(x)\cdot sin(x)}}{\frac{sin^2(x) + cos^2(x)}{cos(x)\cdot sin(x)}}\)

\(\displaystyle = \ \frac{sin^2(x) - cos^2(x)}{cos(x)\cdot sin(x)} \cdot \frac{cos(x)\cdot sin(x)}{sin^2(x) + cos^2(x)}\)

\(\displaystyle = \ sin^2(x) - cos^2(x)\)