I don't understand,please be clear!
Prove that each equation is an identity.
I tried to do the problems, but I am stuck.
1. cos^4 t-sin^4 t=1-2sin^2 t
2. 1/cos s= csc^2 s - csc s cot s
3. (cos x/ sec x -1)- (cos x/ tan^2x)=cot^2 x
4. sin^3 z cos^2 z= sin^3 z - sin^5 z
Sure! I'll break down the steps for each equation to help you understand the proofs.
1. To prove cos^4 t - sin^4 t = 1 - 2sin^2 t:
First, let's use the identity: cos^2 t - sin^2 t = 1.
Rearranging this equation, we have: cos^2 t = 1 - sin^2 t.
Now, let's square both sides of the equation cos^2 t = 1 - sin^2 t:
(cos^2 t)^2 = (1 - sin^2 t)^2.
Expanding this equation, we get: cos^4 t = 1 - 2sin^2 t + sin^4 t.
Therefore, we can substitute cos^4 t = 1 - 2sin^2 t + sin^4 t into the original equation:
cos^4 t - sin^4 t = 1 - 2sin^2 t + sin^4 t - sin^4 t.
Simplifying this equation, we have: cos^4 t - sin^4 t = 1 - 2sin^2 t.
Hence, the equation cos^4 t - sin^4 t = 1 - 2sin^2 t is proven to be an identity.
2. To prove 1/cos s = csc^2 s - csc s cot s:
First, let's write the right-hand side with sine and cosine functions only.
Using the reciprocal identities, we can rewrite csc s and cot s as 1/sin s and cos s/sin s respectively.
Therefore, the right-hand side becomes: csc^2 s - csc s cot s = (1/sin^2 s) - (cos s/sin^2 s).
Combining the fractions, we have: (1 - cos s)/sin^2 s.
Now, let's simplify the left-hand side (1/cos s):
To rationalize the denominator, we multiply the fraction by (cos s)/(cos s):
(1/cos s) * (cos s)/(cos s) = cos s / (cos^2 s).
Using the identity: 1 + tan^2 s = sec^2 s, we can rewrite the denominator:
cos s / (1 + tan^2 s).
Next, we can rewrite tan^2 s as sin^2 s / cos^2 s:
cos s / (1 + sin^2 s / cos^2 s).
To simplify further, we can multiply the fraction by (cos^2 s) / (cos^2 s):
(cos s * cos^2 s) / (1 * cos^2 s + sin^2 s).
Simplifying the numerator and the denominator, we get:
cos^3 s / (cos^2 s + sin^2 s).
Since cos^2 s + sin^2 s is equal to 1, the denominator becomes 1:
cos^3 s / 1 = cos^3 s.
Hence, the left-hand side (1/cos s) is equal to cos^3 s, which also matches the right-hand side (1 - cos s)/sin^2 s.
Therefore,1/cos s = csc^2 s - csc s cot s is an identity.
3. To prove (cos x/sec x - 1) - (cos x/tan^2 x) = cot^2 x:
First, let's rearrange the terms on the left-hand side of the equation:
cos x / sec x - cos x / tan^2 x - 1.
Using the identities sec x = 1/cos x and tan x = sin x / cos x, we can rewrite the equation:
cos x / (1/cos x) - cos x / (sin^2 x / cos^2 x) - 1.
Simplifying the expression, we get:
cos^2 x - cos^3 x / sin^2 x - 1.
Next, we can rewrite cos^2 x as 1 - sin^2 x:
1 - sin^2 x - cos^3 x / sin^2 x - 1.
Canceling out the common terms of -1 and 1, we have:
-sin^2 x - cos^3 x / sin^2 x.
To simplify further, we can divide both numerator and denominator by sin^2 x:
(- sin^2 x / sin^2 x) - (cos^3 x / sin^2 x) = -1 - (cos^3 x / sin^2 x).
Now, using the identity cot^2 x = 1 + csc^2 x, we can rewrite -1 as (- 1 sin^2 x / sin^2 x):
(- sin^2 x - (cos^3 x / sin^2 x)) = - (sin^2 x + cos^3 x / sin^2 x).
This equation matches the right-hand side, cot^2 x.
Hence, (cos x / sec x - 1) - (cos x / tan^2 x) = cot^2 x is an identity.
4. To prove sin^3 z cos^2 z = sin^3 z - sin^5 z:
Let's factor out sin^3 z from the left-hand side of the equation:
sin^3 z (cos^2 z - 1).
Using the identity: 1 - sin^2 z = cos^2 z, we can substitute it into the equation:
sin^3 z (1 - sin^2 z - 1).
Simplifying further, we get:
sin^3 z (- sin^2 z).
Now, we can factor out -sin^2 z from this equation:
- sin^2 z (sin^3 z).
Since -sin^2 z is equal to -1 * sin^2 z, we can rewrite the equation as:
-1 * sin^2 z * sin^3 z.
Multiplying sin^2 z and sin^3 z, we get sin^5 z.
Therefore, -1 * sin^2 z * sin^3 z = -sin^2 z (sin^3 z) = sin^3 z - sin^5 z.
Hence, sin^3 z cos^2 z = sin^3 z - sin^5 z is an identity.
Sure, I'll be glad to help you understand how to prove that each equation is an identity. Here's an explanation for each equation:
1. To prove that cos^4 t - sin^4 t = 1 - 2sin^2 t is an identity, you can start by factoring the left side of the equation using the difference of squares formula. The difference of squares formula states that a^2 - b^2 = (a + b)(a - b). Applying this formula, you can rewrite cos^4 t - sin^4 t as (cos^2 t + sin^2 t)(cos^2 t - sin^2 t). Since cos^2 t + sin^2 t equals 1 (from the Pythagorean identity), you can simplify the expression to (1)(cos^2 t - sin^2 t). Using the difference of squares formula again, you can further simplify it to (1)(cos t + sin t)(cos t - sin t). Finally, you can apply the Pythagorean identity once more to simplify cos t + sin t to 1 and cos t - sin t to cos 2t. Therefore, the final expression is 1(cos 2t) = 1 - 2sin^2 t, which proves that the equation is an identity.
2. To prove that 1/cos s = csc^2 s - csc s cot s is an identity, you can start by rewriting the right side of the equation using trigonometric identities. The csc s can be written as 1/sin s, and cot s can be written as cos s/sin s. Substituting these values in, the equation becomes 1/cos s = (1/sin^2 s) - (1/sin s)(cos s/sin s). To simplify this further, you can combine the fractions on the right side by finding a common denominator. The common denominator is sin^2 s, so the equation becomes 1/cos s = (1 - cos s)/sin^2 s. Then, you can multiply both sides of the equation by cos s to get (1 - cos s)/cos s = sin^2 s. You can further simplify the left side by dividing both the numerator and the denominator by cos s, which gives you 1/cos s - 1 = sin^2 s. Finally, using the reciprocal identity 1/cos s = sec s, the equation becomes sec s - 1 = sin^2 s, proving that it is an identity.
3. To prove that (cos x/sec x - 1) - (cos x/tan^2 x) = cot^2 x is an identity, you can start by simplifying the left side of the equation. The expression cos x/sec x is equivalent to cos x/(1/cos x), which simplifies to cos^2 x. The expression cos x/tan^2 x can be written as (cos x)/(sin^2 x/cos^2 x), which simplifies to cos^3 x/sin^2 x. Applying these simplifications, the equation becomes cos^2 x - (cos^3 x/sin^2 x) = cot^2 x. To simplify this further, you can multiply the first term by sin^2 x/sin^2 x and the second term by cos^2 x/cos^2 x to get (cos^2 x * sin^2 x)/(sin^2 x) - (cos^3 x * cos^2 x)/(sin^2 x * cos^2 x) = cot^2 x. This simplifies to sin^2 x - cos^5 x = cot^2 x. Finally, using the reciprocal identity 1/cot x = tan x, the equation becomes sin^2 x - cos^5 x = (1/tan x)^2, which is equivalent to cot^2 x, proving the identity.
4. To prove that sin^3 z cos^2 z = sin^3 z - sin^5 z is an identity, you can start by factoring out sin^3 z from both terms on the right side of the equation. This gives you sin^3 z (1 - sin^2 z). Since sin^2 z = 1 - cos^2 z (from the pythagorean identity), you can substitute this value in to get sin^3 z (1 - (1 - cos^2 z)). This simplifies to sin^3 z (cos^2 z). Comparing this with the left side of the equation, you can see that both sides are equal, proving the identity.
I hope this explanation helps you better understand how to approach proving trigonometric equations as identities. If you have any more questions or need further clarification, feel free to ask!