Thread: Identities in Terms of Trigonometry

1. Identities in Terms of Trigonometry

I have two questions below.

Establish each identity.

(1) cos(x)/[1 - tan(x)] + sin(x)/[1 - cot(x)] = sin(x) + cos(x)

This is my set up for question 1:

Let tan(x) = sin(x)/cos(x)

Let cot(x) = cos(x)/sin(x)

After plugging and simplifying the complex fractions, I get this on the left side:

cos^2 (x)/[cos(x) - sin(x)] + sin^2 (x)/[sin(x) - cos(x)]

Where do I go from there?

Question 2:

3 sin^2 (x) + 4 cos^2 (x) = 3 + cos^2 (x)

I know that sin^2 (x) = 1 - cos^2 (x)

This is my set up:

3[1 - cos^2 (x)] + 4 cos^2 (x) = 3 + cos^2 (x)

I then use the distributive rule on the left side and get this:

3 - 3cos^2 (x) + 4 cos^2 (x) = 3 + cos^2 (x)

Where do I go from there?

Thanks

2. Originally Posted by magentarita
I have two questions below.

Establish each identity.

(1) cos(x)/[1 - tan(x)] + sin(x)/[1 - cot(x)] = sin(x) + cos(x)

This is my set up for question 1:

Let tan(x) = sin(x)/cos(x)

Let cot(x) = cos(x)/sin(x)

After plugging and simplifying the complex fractions, I get this on the left side:

cos^2 (x)/[cos(x) - sin(x)] + sin^2 (x)/[sin(x) - cos(x)]

Where do I go from there?
Get it all over the common denominator:

$\frac{\cos^2 x - \sin^2 x}{\cos x - \sin x}$.

Factorise the numerator (difference of two squares). Cancel the common factor. Viola. You're left with the RHS. Q.E.D.

3. 1st problem

$\frac{cosx}{1-tanx} +\frac{sinx}{1-cotx} = sinx + cosx$

$\frac{cosx}{1-\frac{sinx}{cosx}} + \frac{sinx}{1-\frac{cosx}{sinx}}$

$\frac {cos^2x}{cosx- sinx} + \frac{-sin^2x}{cosx - sinx}$

$\frac {cos^2x - sin^2x}{cosx - sinx}$

$\frac {(cosx - sinx)(cosx + sinx)}{cosx - sinx}$

$cosx + sinx$

oops I was to slow again

4. Originally Posted by magentarita
I have two questions below.

Establish each identity.

[snip]

Question 2:

3 sin^2 (x) + 4 cos^2 (x) = 3 + cos^2 (x)

I know that sin^2 (x) = 1 - cos^2 (x)

This is my set up:

Mr F edits:

LHS = 3[1 - cos^2 (x)] + 4 cos^2 (x)

I then use the distributive rule on the left side and get this:

= 3 - 3cos^2 (x) + 4 cos^2 (x) = 3 + cos^2 (x)

= RHS.

Where do I go from there? Mr F says: You write Q.E.D. and move on to the next question.

Thanks
..

5. I thank you both

Thanks a lot. I am learning so much math here.

Math has now become my passion thanks to this site.