Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.
(sin x + cos x) ^2
a. 1+2sinxcosx
b. sec^2x−tan^2x+2cosxsinx
c.sec x + 2 sin x/sec x
d. sin^2x+cos^2x
e. 1+2cos (pi / 2 - x) cos x
you know that sin^2 x + cos^2 x = 1, so when you expand you have
sin^2 x + 2 sinx cosx + cos^2 x
recall also that sec^2 x = 1 + tan^2 x
cos(pi/2-x) = sin(x)
So, what do you think?
To simplify the expression (sin x + cos x)^2, we can use the distributive property of exponents which states that (a + b)^2 = a^2 + 2ab + b^2. Let's apply this identity to the given expression.
Expanding (sin x + cos x)^2:
= (sin x)^2 + 2(sin x)(cos x) + (cos x)^2
= sin^2x + 2sinxcosx + cos^2x
Now let's simplify each option and determine which one is not equivalent.
a. 1 + 2sinxcosx
Here, we have 1 + 2sin xcos x, which is the same as the expanded form of (sin x + cos x)^2. Therefore, option a is equivalent to the given expression.
b. sec^2x − tan^2x + 2cosxsinx
To simplify this option, we can use the fundamental trigonometric identities. Recall that sec^2x = 1 + tan^2x and tan^2x + 1 = sec^2x.
Using these identities, we can rewrite option b as:
= (1 + tan^2x) - tan^2x + 2cosxsinx
= 1 + tan^2x - tan^2x + 2cosxsinx
= 1 + 2cosxsinx
Option b simplifies to 1 + 2cosxsinx, which is also equivalent to the given expression. Therefore, option b is not the answer.
c. sec x + 2 sin x/sec x
To simplify this option, we can simplify the expression sec x + 2 sin x/sec x separately.
sec x = 1/cos x
2 sin x/sec x = 2 sin x/(1/cos x) = 2 sin x cos x
Therefore, option c simplifies to sin x + 2sin x cos x, which is not equivalent to the given expression. Therefore, option c is the correct answer.
d. sin^2x + cos^2x
Here, we have the well-known trigonometric identity sin^2x + cos^2x = 1. This means that option d is equivalent to the given expression.
e. 1 + 2cos (pi / 2 - x) cos x
To simplify this option, we can manipulate the cosine function using the angle difference identity.
cos (pi / 2 - x) = sin x
Substituting this into option e, we get:
1 + 2cos (pi / 2 - x) cos x = 1 + 2sinx cos x
Therefore, option e is equivalent to the given expression.
To summarize, options a, b, d, and e are all equivalent to (sin x + cos x)^2, while option c, sec x + 2 sin x/sec x, is not equivalent.
To simplify the expression (sin x + cos x)^2, we can multiply it out and then apply the fundamental trigonometric identities.
(sin x + cos x)^2 = (sin x + cos x)(sin x + cos x)
= sin^2 x + sin x cos x + cos x sin x + cos^2 x
= sin^2 x + 2sin x cos x + cos^2 x
Now, let's compare this simplified expression with the given options to determine which one is not equivalent.
Option a: 1 + 2sin x cos x
This is equivalent to the simplified expression, so it is a valid choice.
Option b: sec^2 x − tan^2 x + 2cos x sin x
Using the fundamental identities, sec^2 x = 1 + tan^2 x, so we can rewrite this as:
1 + tan^2 x − tan^2 x + 2cos x sin x
Simplifying further, we have:
1 + 2cos x sin x
This is also equivalent to the simplified expression, so it is a valid choice.
Option c: sec x + 2 sin x/sec x
Using the fundamental identity, sec x = 1/cos x, we can rewrite this expression as:
1/cos x + 2 sin x/(1/cos x)
Simplifying further, we get:
1/cos x + 2 sin x cos x
This is equivalent to the simplified expression, so it is a valid choice.
Option d: sin^2 x + cos^2 x
This is the Pythagorean identity, and it is equivalent to the simplified expression. Hence, it is a valid choice.
Option e: 1 + 2cos (pi/2 - x) cos x
We can simplify this expression using the cosine angle sum identity:
cos (pi/2 - x) = sin x
Substituting this into the expression, we have:
1 + 2cos x sin x
This is not equivalent to the simplified expression, so the correct answer is option e.
Therefore, the expression that is not equivalent to (sin x + cos x)^2 is e. 1 + 2cos (pi/2 - x) cos x.