Need help proving some trig identities

In summary, for the first problem, the identity 1 + sec^(2)xsin^(2)x = sec^(2)x can be proven by replacing sec x with its cosine definition and applying a Pythagorean identity to the numerator. For the second problem, the identity sinx/1-cosx + sinx/1+cosx = 2cscx can be proven by combining similar fractions on the left side and using the reciprocal identity for cosecant.
  • #1
Confusedbrah
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Proving identities is a pain! Thanks in advance, guys!

Homework Statement



1. 1 + sec^(2)xsin^(2)x = sec^(2)x

2. sinx/1-cosx + sinx/1+cosx = 2cscx

Homework Equations


The Attempt at a Solution



For the first problem, this is the best I got:

1 + sec^(2)x(1-cos(2)x)

For the second problem, I added the fractions together and got:

(sinx + sinxcosx + sinx - sinxcosx) / ((1-cosx) (1-cosx)) =

(2sinx) / (1-cosx) (1+cosx) =

(2sinx) / (1-cos^(2)x) =

(2sinx) / (sin^(2)x)
 
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  • #2
Welcome to PF

For the first one, go back to basics and replace sec with its cos definition ie sec(x)=1/cos(x) and it should be apparent.

For the second one, try combining the fractions on the left and also replace csc(x) with its sin(x) definition.
 
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  • #3
jedishrfu said:
Welcome to PF

For the first one, go back to basics and replace sec with its cos definition ie sec(x)=1/cos(x) and it should be apparent.

For the second one, try combining the fractions on the left and also replace csc(x) with its sin(x) definition.

Hmmm I kinda see what you're saying. I started all over with the first one and this is what I got, can anyone check if I did it correctly? It proves to be correct, at least to me.

1 + (1/cos^(2)x)(sin^(2))x) = I then multiplied the two fractions, giving me:

1 + (sin^(2)x)/(cos^(2)x) = now I combine the fractions, giving me:

(cos^(2)x + sin^(2)x)/(cos^(2)x) = now I apply a Pythagorean identity to the numerator:

(1)/(cos^(2)x) = sec^(2)x


I am still stumped on the second one. Are you saying combine fractions from the initial equation or the one I worked on and got? Thanks
 
  • #4
Combine the LHS of number 2. Do you see why that is a good idea?
 
  • #5
verty said:
Combine the LHS of number 2. Do you see why that is a good idea?

I think I got it now. For number 2, I separated the two fractions from the left side of the initial equation and got this:

(sinx)/(1) - (sinx)/(cosx) + (sinx)/(1) +(sinx)/(cosx) = then I combined similar terms and got this:

(sinx)/(1) + (sinx)/(1) = now I flipped the fractions and got this:

(1)/(sinx) + (1)/(sinx) = (2)/(sinx) = 2cscx

Can anyone check if I did it correctly? Thanks
 

FAQ: Need help proving some trig identities

1. What are trig identities?

Trig identities are equations that involve trigonometric functions (such as sine, cosine, and tangent) and are true for all values of the variables involved.

2. Why is it important to prove trig identities?

Proving trig identities is important because it helps us understand the relationships between different trigonometric functions and allows us to simplify complex expressions.

3. What are some common techniques for proving trig identities?

Some common techniques for proving trig identities include using basic trigonometric identities, manipulating the expressions using algebraic techniques, and using the properties of even and odd functions.

4. How can I check if my proof for a trig identity is correct?

The best way to check if your proof is correct is to work through the steps backwards and see if you end up with the original expression. You can also use a calculator to evaluate both sides of the identity for different values of the variables.

5. Are there any tips for proving trig identities more efficiently?

One tip for proving trig identities more efficiently is to start with the more complicated side of the equation and simplify it using known identities. It's also helpful to have a good understanding of the basic trigonometric identities and their properties.

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