Westonci.ca is the premier destination for reliable answers to your questions, brought to you by a community of experts. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To solve the given equation [tex]\(\frac{(1+\tan x)^2}{\sec x} = \sec x + 2 \sin x\)[/tex], we will simplify both sides using trigonometric identities.
### Step 1: Simplify the left-hand side (LHS)
We begin with:
[tex]\[ \frac{(1 + \tan x)^2}{\sec x} \][/tex]
First, recall the trigonometric identities:
[tex]\[ \tan x = \frac{\sin x}{\cos x} \][/tex]
[tex]\[ \sec x = \frac{1}{\cos x} \][/tex]
Using these identities, we can rewrite [tex]\(1 + \tan x\)[/tex] as:
[tex]\[ 1 + \tan x = 1 + \frac{\sin x}{\cos x} = \frac{\cos x + \sin x}{\cos x} \][/tex]
Then, square it:
[tex]\[ (1 + \tan x)^2 = \left( \frac{\cos x + \sin x}{\cos x} \right)^2 = \frac{(\cos x + \sin x)^2}{\cos^2 x} \][/tex]
Now, dividing by [tex]\(\sec x\)[/tex]:
[tex]\[ \frac{(1 + \tan x)^2}{\sec x} = \frac{(\cos x + \sin x)^2}{\cos^2 x} \cdot \cos x = \frac{(\cos x + \sin x)^2}{\cos x} \][/tex]
### Step 2: Expand and simplify
Expand [tex]\((\cos x + \sin x)^2\)[/tex]:
[tex]\[ (\cos x + \sin x)^2 = \cos^2 x + 2 \cos x \sin x + \sin^2 x \][/tex]
Using the Pythagorean identity [tex]\(\cos^2 x + \sin^2 x = 1\)[/tex], we have:
[tex]\[ \cos^2 x + \sin^2 x = 1 \][/tex]
Substitute back into our expression:
[tex]\[ \frac{(1 + \tan x)^2}{\sec x} = \frac{1 + 2 \cos x \sin x}{\cos x} \][/tex]
### Step 3: Simplify further
Note that:
[tex]\[ 2 \cos x \sin x = \sin(2x) \][/tex]
However, it may not directly help us here. Instead, we proceed by simply substituting and separating terms in the numerator:
[tex]\[ \frac{(1 + \tan x)^2}{\sec x} = \frac{1}{\cos x} + \frac{2 \cos x \sin x}{\cos x} \][/tex]
[tex]\[ \frac{(1 + \tan x)^2}{\sec x} = \sec x + 2 \sin x \][/tex]
### Step 4: Compare with the right-hand side (RHS)
We already have:
[tex]\[ \sec x + 2 \sin x \][/tex]
So our simplified LHS becomes:
[tex]\[ \sec x + 2 \sin x \][/tex]
Which exactly matches the RHS:
[tex]\[ \sec x + 2 \sin x \][/tex]
Thus, we have verified that:
[tex]\[ \frac{(1+\tan x)^2}{\sec x} = \sec x + 2 \sin x \][/tex]
Hence, the given equation is an identity and is satisfied for all [tex]\(x\)[/tex] where the trigonometric functions are defined.
### Step 1: Simplify the left-hand side (LHS)
We begin with:
[tex]\[ \frac{(1 + \tan x)^2}{\sec x} \][/tex]
First, recall the trigonometric identities:
[tex]\[ \tan x = \frac{\sin x}{\cos x} \][/tex]
[tex]\[ \sec x = \frac{1}{\cos x} \][/tex]
Using these identities, we can rewrite [tex]\(1 + \tan x\)[/tex] as:
[tex]\[ 1 + \tan x = 1 + \frac{\sin x}{\cos x} = \frac{\cos x + \sin x}{\cos x} \][/tex]
Then, square it:
[tex]\[ (1 + \tan x)^2 = \left( \frac{\cos x + \sin x}{\cos x} \right)^2 = \frac{(\cos x + \sin x)^2}{\cos^2 x} \][/tex]
Now, dividing by [tex]\(\sec x\)[/tex]:
[tex]\[ \frac{(1 + \tan x)^2}{\sec x} = \frac{(\cos x + \sin x)^2}{\cos^2 x} \cdot \cos x = \frac{(\cos x + \sin x)^2}{\cos x} \][/tex]
### Step 2: Expand and simplify
Expand [tex]\((\cos x + \sin x)^2\)[/tex]:
[tex]\[ (\cos x + \sin x)^2 = \cos^2 x + 2 \cos x \sin x + \sin^2 x \][/tex]
Using the Pythagorean identity [tex]\(\cos^2 x + \sin^2 x = 1\)[/tex], we have:
[tex]\[ \cos^2 x + \sin^2 x = 1 \][/tex]
Substitute back into our expression:
[tex]\[ \frac{(1 + \tan x)^2}{\sec x} = \frac{1 + 2 \cos x \sin x}{\cos x} \][/tex]
### Step 3: Simplify further
Note that:
[tex]\[ 2 \cos x \sin x = \sin(2x) \][/tex]
However, it may not directly help us here. Instead, we proceed by simply substituting and separating terms in the numerator:
[tex]\[ \frac{(1 + \tan x)^2}{\sec x} = \frac{1}{\cos x} + \frac{2 \cos x \sin x}{\cos x} \][/tex]
[tex]\[ \frac{(1 + \tan x)^2}{\sec x} = \sec x + 2 \sin x \][/tex]
### Step 4: Compare with the right-hand side (RHS)
We already have:
[tex]\[ \sec x + 2 \sin x \][/tex]
So our simplified LHS becomes:
[tex]\[ \sec x + 2 \sin x \][/tex]
Which exactly matches the RHS:
[tex]\[ \sec x + 2 \sin x \][/tex]
Thus, we have verified that:
[tex]\[ \frac{(1+\tan x)^2}{\sec x} = \sec x + 2 \sin x \][/tex]
Hence, the given equation is an identity and is satisfied for all [tex]\(x\)[/tex] where the trigonometric functions are defined.
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.