Answer :
Of course! Let's solve and verify the trigonometric identity step-by-step:
Given identity to verify:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = 2 \cos 2\theta + 1 \][/tex]
### Step 1: Simplify the left-hand side
First, we start by expanding the left-hand side of the given equation.
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) \][/tex]
We use the difference of squares formula [tex]\( (a+b)(a-b) = a^2 - b^2 \)[/tex]:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = (2 \cos \theta)^2 - 1^2 \][/tex]
Now, compute each part:
[tex]\[ (2 \cos \theta)^2 = 4 \cos^2 \theta \][/tex]
[tex]\[ 1^2 = 1 \][/tex]
So, substituting back, we have:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = 4 \cos^2 \theta - 1 \][/tex]
Thus, the left-hand side simplifies to:
[tex]\[ 4 \cos^2 \theta - 1 \][/tex]
### Step 2: Simplify the right-hand side
Now let's simplify the right-hand side of the original equation:
[tex]\[ 2 \cos 2\theta + 1 \][/tex]
### Step 3: Use a known trigonometric identity
We will now use a trigonometric identity to relate [tex]\(\cos(2\theta)\)[/tex] directly with [tex]\(\cos(\theta)\)[/tex]. One of the double angle identities for cosine is:
[tex]\[ \cos(2\theta) = 2 \cos^2 \theta - 1 \][/tex]
Therefore, substitute [tex]\(\cos(2\theta)\)[/tex] in the right-hand side expression:
[tex]\[ 2 \cos(2\theta) + 1 = 2(2 \cos^2 \theta - 1) + 1 \][/tex]
Simplify inside the parentheses first:
[tex]\[ = 2 \cdot 2 \cos^2 \theta - 2 \cdot 1 + 1 \][/tex]
[tex]\[ = 4 \cos^2 \theta - 2 + 1 \][/tex]
[tex]\[ = 4 \cos^2 \theta - 1 \][/tex]
### Step 4: Compare both sides
Now, compare the simplified forms of both sides of the original equation:
Left-hand side:
[tex]\[ 4 \cos^2 \theta - 1 \][/tex]
Right-hand side:
[tex]\[ 4 \cos^2 \theta - 1 \][/tex]
Since the simplified forms of both sides are equal, we have verified the identity is true.
### Conclusion
Therefore, the identity:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = 2 \cos 2\theta + 1 \][/tex]
is verified to be true.
Given identity to verify:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = 2 \cos 2\theta + 1 \][/tex]
### Step 1: Simplify the left-hand side
First, we start by expanding the left-hand side of the given equation.
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) \][/tex]
We use the difference of squares formula [tex]\( (a+b)(a-b) = a^2 - b^2 \)[/tex]:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = (2 \cos \theta)^2 - 1^2 \][/tex]
Now, compute each part:
[tex]\[ (2 \cos \theta)^2 = 4 \cos^2 \theta \][/tex]
[tex]\[ 1^2 = 1 \][/tex]
So, substituting back, we have:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = 4 \cos^2 \theta - 1 \][/tex]
Thus, the left-hand side simplifies to:
[tex]\[ 4 \cos^2 \theta - 1 \][/tex]
### Step 2: Simplify the right-hand side
Now let's simplify the right-hand side of the original equation:
[tex]\[ 2 \cos 2\theta + 1 \][/tex]
### Step 3: Use a known trigonometric identity
We will now use a trigonometric identity to relate [tex]\(\cos(2\theta)\)[/tex] directly with [tex]\(\cos(\theta)\)[/tex]. One of the double angle identities for cosine is:
[tex]\[ \cos(2\theta) = 2 \cos^2 \theta - 1 \][/tex]
Therefore, substitute [tex]\(\cos(2\theta)\)[/tex] in the right-hand side expression:
[tex]\[ 2 \cos(2\theta) + 1 = 2(2 \cos^2 \theta - 1) + 1 \][/tex]
Simplify inside the parentheses first:
[tex]\[ = 2 \cdot 2 \cos^2 \theta - 2 \cdot 1 + 1 \][/tex]
[tex]\[ = 4 \cos^2 \theta - 2 + 1 \][/tex]
[tex]\[ = 4 \cos^2 \theta - 1 \][/tex]
### Step 4: Compare both sides
Now, compare the simplified forms of both sides of the original equation:
Left-hand side:
[tex]\[ 4 \cos^2 \theta - 1 \][/tex]
Right-hand side:
[tex]\[ 4 \cos^2 \theta - 1 \][/tex]
Since the simplified forms of both sides are equal, we have verified the identity is true.
### Conclusion
Therefore, the identity:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = 2 \cos 2\theta + 1 \][/tex]
is verified to be true.
For the first one 3x= 6x-2 the answer is 1.5 since the 6- is positive you want to subtract it with 3x since it’s the only other number with x, you subtract 3x-6x and get -3x, now the equation is -3x=-2, Divide the -2 with -3x to get rid of the -2, -2/-2, -3x/-2
X=1.5
X=1.5
