Are any of these three problems identities?
1. Cos^2x-sin^2x=1-2sin^2x
2. Sinxsecx=cosxcscx
3. Sec^4x-tan^4/sec^2x=1+sin^2x
If so, how can you conclude that any of them are identities?
Yes. You can prove an identity by rearranging one side of the equation to match the other.
For example:
1.
cos^2(x) - sin^2(x) = 1 - sin^2(x) - sin^2(x) because cos^2(x) + sin^2(x) = 1
= 1 - 2 sin^2(x)
The second one is not an identity. You can prove this by substituting the appropriate sin/cos functions for sec(x) and csc(x).
In order to determine if any of these three problems are identities, we need to simplify both sides of the equation and show that they are equivalent. If we can transform both sides of the equation into the same expression, then it is an identity.
1. To determine if Cos^2x - sin^2x = 1 - 2sin^2x is an identity, we first simplify both sides of the equation:
- Starting with the left side, we can use the trigonometric identity cos^2x - sin^2x = cos(2x).
- On the right side, we can simplify 1 - 2sin^2x as cos^2x, using the trigonometric identity sin^2x + cos^2x = 1.
- Now both sides of the equation are simplified to cos(2x), which means that the equation holds true for all values of x. Therefore, it is an identity.
2. To determine if sinxsecx = cosxcscx is an identity, we simplify both sides of the equation:
- On the left side, secx is equal to 1/cosx, so we can rewrite sinxsecx as sinx(1/cosx) which simplifies to sinx/cosx, or tanx using the trigonometric identity sinx/cosx = tanx.
- On the right side, cscx is equal to 1/sinx, so we can rewrite cosxcscx as cosx(1/sinx) which simplifies to cosx/sinx, or cotx using the trigonometric identity cosx/sinx = cotx.
- Now both sides of the equation are simplified to tanx = cotx. However, this is not true for all values of x, so the equation is not an identity.
3. To determine if sec^4x - tan^4x/sec^2x = 1 + sin^2x is an identity, we simplify both sides of the equation:
- On the left side, we can simplify sec^4x as (1/cosx)^4 = 1/cos^4x, and tan^4x as (sinx/cosx)^4 = sin^4x/cos^4x.
- Dividing sin^4x/cos^4x by 1/cos^2x gives us sin^4x/cos^2x.
- The equation now becomes 1/cos^4x - sin^4x/cos^2x = 1 + sin^2x.
- We can simplify 1/cos^4x - sin^4x/cos^2x by finding a common denominator, which results in (1 - sin^4x)/cos^4x.
- Now our equation is (1 - sin^4x)/cos^4x = 1 + sin^2x.
- Multiplying both sides by cos^4x yields 1 - sin^4x = cos^4x + sin^2x * cos^4x.
- Simplifying further, we get 1 - sin^4x = cos^4x + sin^2x * (1 - sin^2x).
- Expanding the expression in the parentheses gives 1 - sin^4x = cos^4x + sin^2x - sin^4x.
- The terms -sin^4x cancel out, leaving us with 1 = cos^4x + sin^2x.
- Since this equation holds true for all values of x, it is an identity.
Therefore, out of the three given problems, the first and the third are identities, while the second is not.