How do I do these problems?
Verify the identity.
a= alpha, b=beta, t= theta
1. (1 + sin a) (1 - sin a)= cos^2a
2. cos^2b - sin^2b = 2cos^2b - 1
3. sin^2a - sin^4a = cos^2a - cos^4a
4. (csc^2 t / cot t) = csc t sec t
5. (cot^2 t / csc t) = csc t = sin t
Learn your identites well in order to prove these.
1. (1+sina) (1-sina)
= 1-sina+sina-sin^2a
= 1-sin^2a
= cos^2a (according to identity)
2. cos^2b-sin^2b
you know that sin^2b = 1-cos^2b, so:
cos^2b-(1-cos^2b)
=cos^2b-1+cos^2b
=2cos^2b-1
3. sin^2a - sin^4a=
= sin^2a- (sin^2a)(sin^2a)
=(1-cos^2a)-(1-cos^2a)(1-cos^2a)
=1-cos^2a- (1-2cos^2a+cos^4a)
=1-cos^2a-1+2cos^2a-cos^4a
=cos^2a-cos^4a
I hope you can do the rest by yourself ;)
1-sin^2B
------------- = csc^2B-sec^2B
sin^2Bcos^2B
To verify each identity step-by-step, let's start with the first one:
1. (1 + sin a) (1 - sin a) = cos^2a
Step 1: Expand the left side of the equation.
(1 + sin a) (1 - sin a) = 1 - sin^2a
Step 2: Use the Pythagorean Identity sin^2a + cos^2a = 1.
1 - sin^2a = cos^2a
Therefore, (1 + sin a) (1 - sin a) = cos^2a is verified.
Let's move on to the second identity:
2. cos^2b - sin^2b = 2cos^2b - 1
Step 1: Rearrange the right side of the equation.
2cos^2b - 1 = 2(cos^2b - 1/2)
Step 2: Use the Pythagorean Identity sin^2b = 1 - cos^2b.
2(cos^2b - 1/2) = 2(1 - sin^2b - 1/2)
Step 3: Simplify the equation.
2(1 - sin^2b - 1/2) = cos^2b - sin^2b
Therefore, cos^2b - sin^2b = 2cos^2b - 1 is verified.
Moving on to the third identity:
3. sin^2a - sin^4a = cos^2a - cos^4a
Step 1: Rearrange the left side of the equation.
sin^2a - sin^4a = sin^2a (1 - sin^2a)
Step 2: Use the Pythagorean Identity cos^2a = 1 - sin^2a.
sin^2a (1 - sin^2a) = sin^2a (cos^2a)
Step 3: Distribute sin^2a to cos^2a.
sin^2a (cos^2a) = cos^2a sin^2a
Step 4: Rearrange the right side of the equation.
cos^2a sin^2a = cos^2a (1 - cos^2a)
Step 5: Use the Pythagorean Identity sin^2a = 1 - cos^2a.
cos^2a (1 - cos^2a) = cos^2a - cos^4a
Therefore, sin^2a - sin^4a = cos^2a - cos^4a is verified.
Let's move on to the fourth identity:
4. (csc^2t / cot t) = csc t sec t
Step 1: Convert csc t to its reciprocal form, 1 / sin t.
(csc^2t / cot t) = (1 / (sin t)^2) / (cos t / sin t)
Step 2: Invert the second fraction and multiply.
(1 / (sin t)^2) / (cos t / sin t) = (1 / (sin t)^2) x (sin t / cos t)
Step 3: Simplify the equation using the reciprocal and quotient identities.
(1 / (sin t)^2) x (sin t / cos t) = (1 / cos t) = sec t
Therefore, (csc^2t / cot t) = csc t sec t is verified.
Finally, let's look at the fifth identity:
5. (cot^2t / csc t) = csc t = sin t
Step 1: Convert cot t to its reciprocal form, 1 / tan t.
(cot^2t / csc t) = ((1 / tan t)^2) / (1 / sin t)
Step 2: Invert the second fraction and multiply.
((1 / tan t)^2) / (1 / sin t) = ((1 / tan t)^2) x (sin t)
Step 3: Simplify the equation using the reciprocal identity.
((1 / tan t)^2) x (sin t) = (cos^2t / sin t) x (sin t)
Step 4: Simplify further by canceling out sin t.
(cos^2t / sin t) x (sin t) = cos^2t
Therefore, (cot^2t / csc t) = csc t = sin t is verified.
I hope this step-by-step explanation helps! Let me know if you have any further questions.
To verify the identities mentioned, we will use the basic trigonometric identities. Let's go through each problem step by step:
1. (1 + sin a) (1 - sin a) = cos^2a
To solve this problem, we will use the Pythagorean identity: sin^2a + cos^2a = 1.
Start with the left-hand side of the equation:
(1 + sin a) (1 - sin a)
Using the difference of squares formula, we get:
(1 - sin^2a)
Using the Pythagorean identity sin^2a + cos^2a = 1, we can rewrite the above expression:
cos^2a
Therefore, the left-hand side is equal to the right-hand side. The identity is verified.
2. cos^2b - sin^2b = 2cos^2b - 1
To solve this problem, we will again use the Pythagorean identity: sin^2b + cos^2b = 1.
Start with the left-hand side of the equation:
cos^2b - sin^2b
Using the Pythagorean identity sin^2b + cos^2b = 1, we can rewrite the above expression:
cos^2b - (1 - cos^2b)
Simplifying further:
cos^2b - 1 + cos^2b
Rearranging the terms:
2cos^2b - 1
Therefore, the left-hand side is equal to the right-hand side. The identity is verified.
3. sin^2a - sin^4a = cos^2a - cos^4a
To solve this problem, we will again use the Pythagorean identity: sin^2a + cos^2a = 1.
Start with the left-hand side of the equation:
sin^2a - sin^4a
Factoring out sin^2a:
sin^2a(1 - sin^2a)
Using the Pythagorean identity sin^2a + cos^2a = 1, we can replace (1 - sin^2a) with cos^2a:
sin^2a(cos^2a)
Rearranging the terms:
cos^2a(sin^2a)
Using the commutative property of multiplication, we get:
cos^2a(sin^2a) = cos^2a(sin^2a)
Therefore, the left-hand side is equal to the right-hand side. The identity is verified.
4. (csc^2t / cot t) = csc t sec t
To solve this problem, we will use the reciprocal and quotient identities.
Start with the left-hand side of the equation:
(csc^2t / cot t)
Using the reciprocal identity:
csc^2t = 1 / sin^2t
Using the quotient identity:
cot t = cos t / sin t
Substituting the above values:
(1 / sin^2t) / (cos t / sin t)
Multiplying by the reciprocal of the denominator:
(1 / sin^2t) * (sin t / cos t)
Simplifying:
1 / (sin t * cos t)
Using the reciprocal identity:
1 / sin t * 1 / cos t
Rearranging the terms:
csc t * sec t
Therefore, the left-hand side is equal to the right-hand side. The identity is verified.
5. (cot^2t / csc t) = csc t = sin t
To solve this problem, we will again use the reciprocal and quotient identities.
Start with the left-hand side of the equation:
(cot^2t / csc t)
Using the reciprocal identity:
cot^2t = cos^2t / sin^2t
Using the reciprocal identity for csc t:
csc t = 1 / sin t
Substituting the above values:
(cos^2t / sin^2t) / (1 / sin t)
Multiplying by the reciprocal of the denominator:
(cos^2t / sin^2t) * (sin t / 1)
Simplifying:
(cos^2t / sin t)
Using the reciprocal identity:
1 / (sin t / cos^2t)
Rearranging the terms:
csc t * sin t
Therefore, the left-hand side is equal to the right-hand side. The identity is verified.