For all values of x for which the expressions are defined, prove the following equation identity

Question

For all values of x for which the expressions are defined, prove the following equation identity

2/sec = 2-2 sin^2 x/ cos x

Answer 1

LS = 2cosx

RS = 2(1 - sin^2 x)/cosx
= 2cos^2 x/cosx
= 2cosx
= LS

Answer 2

To prove the given equation identity, which is:

2/sec(x) = 2 - 2sin^2(x)/cos(x),

we will simplify the left-hand side (LHS) and the right-hand side (RHS) and show that they are equal.

First, let's simplify the LHS:

LHS: 2/sec(x)

Recall that the secant function (sec(x)) is the reciprocal of the cosine function (cos(x)). So, we can rewrite the LHS as:

LHS = 2/cos(x)

Next, let's simplify the RHS:

RHS: 2 - 2sin^2(x)/cos(x)

We can simplify this by combining the two terms on the RHS:

RHS = 2 - 2sin^2(x)/cos(x)

Now, let's find a common denominator for the two terms on the RHS:

RHS = (2 * cos(x))/cos(x) - (2sin^2(x))/cos(x)

Simplifying further, we have:

RHS = (2cos(x) - 2sin^2(x))/cos(x)

Now, we can write the numerator of the RHS as:

2cos(x) - 2sin^2(x) = 2(cos(x) - sin^2(x))

Recall the trigonometric identity: 1 - sin^2(x) = cos^2(x). Therefore, we can rewrite the numerator of the RHS as:

2(cos(x) - sin^2(x)) = 2(cos(x) - (1 - cos^2(x)))

Simplifying:

2(cos(x) - (1 - cos^2(x))) = 2(cos(x) - 1 + cos^2(x))

Now, let's substitute this expression back into the RHS:

RHS = (2(cos(x) - 1 + cos^2(x)))/cos(x)

Now, let's simplify this expression:

RHS = (2cos(x) - 2 + 2cos^2(x))/cos(x)

Combining like terms:

RHS = (2cos^2(x) + 2cos(x) - 2)/cos(x)

Factoring out a 2 from the numerator:

RHS = 2(cos^2(x) + cos(x) - 1)/cos(x)

Now, we can see that the numerator of the RHS is equal to (cos(x) + 1)(cos(x) - 1). So, we can rewrite the RHS as:

RHS = 2(cos(x) + 1)(cos(x) - 1)/cos(x)

Now, let's simplify this further:

RHS = 2(cos(x) + 1)(cos(x) - 1)/cos(x)

= 2(cos(x) + 1)(cos(x)/cos(x) - 1/cos(x))

= 2(cos(x) + 1)(1 - sec(x))

Using the fact that sec(x) = 1/cos(x), we can simplify further:

= 2(cos(x) + 1)(1 - 1/cos(x))

= 2(cos(x) + 1)(sec(x) - 1)

Now, we can see that the RHS is equal to:

2(cos(x) + 1)(sec(x) - 1).

Since we have shown that the LHS is equal to 2(cos(x) + 1)(sec(x) - 1), we can conclude that:

2/sec(x) = 2 - 2sin^2(x)/cos(x).

Thus, the given equation identity is proven.

Answer 3

To prove the given equation identity `2/sec(x) = 2 - (2sin^2(x))/cos(x)`, we need to simplify both sides of the equation separately and demonstrate that they are equal.

Let's start by simplifying the left side of the equation.

sec(x) is the reciprocal of cos(x). Hence, sec(x) = 1/cos(x).

Therefore, the left side of the equation becomes: 2/sec(x) = 2/(1/cos(x)) = 2 * cos(x).

Now, let's simplify the right side of the equation.

We can start by simplifying the expression (2sin^2(x))/cos(x).

First, let's deal with the numerator: sin^2(x).

sin^2(x) can be rewritten as (1 - cos^2(x)). This is using the trigonometric identity sin^2(x) + cos^2(x) = 1.

Substituting, (2sin^2(x))/cos(x) = (2(1 - cos^2(x)))/cos(x).

Expanding the numerator further, (2(1 - cos^2(x)))/cos(x) = (2 - 2cos^2(x))/cos(x).

Now, let's continue simplifying the denominator by dividing each term by cos(x).

(2 - 2cos^2(x))/cos(x) = (2cos(x) - 2cos^2(x))/cos(x) = 2 - 2cos(x).

Now, we have simplified the right side of the equation to 2 - 2cos(x).

Hence, the equation identity 2/sec(x) = 2 - (2sin^2(x))/cos(x) can be rewritten as 2cos(x) = 2 - 2cos(x).

Observe that both sides of the equation are equal, which proves the given identity.