Answer :
To solve the given trigonometric equation:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \operatorname{Sin} 4A \cdot \operatorname{Cosec} A \][/tex]
we need to manipulate and simplify both sides of the equation using known trigonometric identities.
### Step 1: Rewrite the given equation using trigonometric identities:
The given equation is:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \operatorname{Sin} 4A \cdot \operatorname{Cosec} A \][/tex]
We know from trigonometric identities that:
[tex]\[ \operatorname{Cosec} A = \frac{1}{\operatorname{Sin} A} \][/tex]
Substituting [tex]\(\operatorname{Cosec} A\)[/tex] in the equation, we get:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \operatorname{Sin} 4A \cdot \left(\frac{1}{\operatorname{Sin} A}\right) \][/tex]
### Step 2: Simplify the expression on the right side:
The equation now becomes:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \left(\frac{\operatorname{Sin} 4A}{\operatorname{Sin} A}\right) \][/tex]
### Step 3: Apply the identity for [tex]\( \operatorname{Sin} 4A \)[/tex]:
Recall the trigonometric identity for [tex]\( \operatorname{Sin} 4A \)[/tex]:
[tex]\[ \operatorname{Sin} 4A = 2 \operatorname{Sin} 2A \cdot \operatorname{Cos} 2A \][/tex]
Substitute this identity into the equation:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \left(\frac{2 \operatorname{Sin} 2A \cdot \operatorname{Cos} 2A}{\operatorname{Sin} A}\right) \][/tex]
### Step 4: Simplify the expression further:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \cdot \frac{2 \operatorname{Sin} 2A \cdot \operatorname{Cos} 2A}{\operatorname{Sin} A} \][/tex]
This simplifies to:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \cdot \frac{2 \operatorname{Sin} 2A \cdot \operatorname{Cos} 2A}{\operatorname{Sin} A} \][/tex]
Thus, we have the following key steps in solving the equation:
1. Original Equation:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \operatorname{Sin} 4A \cdot \operatorname{Cosec} A \][/tex]
2. Using Trigonometric Identities:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \operatorname{Sin} 4A \cdot \frac{1}{\operatorname{Sin} A} \][/tex]
3. Simplifying the Right Side:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \left(\frac{\operatorname{Sin} 4A}{\operatorname{Sin} A}\right) \][/tex]
4. Applying the Identity for [tex]\( \operatorname{Sin} 4A \)[/tex]:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \cdot 2 \cdot \frac{\operatorname{Sin} 2A \cdot \operatorname{Cos} 2A}{\operatorname{Sin} A} \][/tex]
This brings us to the simplified form of the equation, having manipulated it solely using trigonometric identities and simplification methods.
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \operatorname{Sin} 4A \cdot \operatorname{Cosec} A \][/tex]
we need to manipulate and simplify both sides of the equation using known trigonometric identities.
### Step 1: Rewrite the given equation using trigonometric identities:
The given equation is:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \operatorname{Sin} 4A \cdot \operatorname{Cosec} A \][/tex]
We know from trigonometric identities that:
[tex]\[ \operatorname{Cosec} A = \frac{1}{\operatorname{Sin} A} \][/tex]
Substituting [tex]\(\operatorname{Cosec} A\)[/tex] in the equation, we get:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \operatorname{Sin} 4A \cdot \left(\frac{1}{\operatorname{Sin} A}\right) \][/tex]
### Step 2: Simplify the expression on the right side:
The equation now becomes:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \left(\frac{\operatorname{Sin} 4A}{\operatorname{Sin} A}\right) \][/tex]
### Step 3: Apply the identity for [tex]\( \operatorname{Sin} 4A \)[/tex]:
Recall the trigonometric identity for [tex]\( \operatorname{Sin} 4A \)[/tex]:
[tex]\[ \operatorname{Sin} 4A = 2 \operatorname{Sin} 2A \cdot \operatorname{Cos} 2A \][/tex]
Substitute this identity into the equation:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \left(\frac{2 \operatorname{Sin} 2A \cdot \operatorname{Cos} 2A}{\operatorname{Sin} A}\right) \][/tex]
### Step 4: Simplify the expression further:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \cdot \frac{2 \operatorname{Sin} 2A \cdot \operatorname{Cos} 2A}{\operatorname{Sin} A} \][/tex]
This simplifies to:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \cdot \frac{2 \operatorname{Sin} 2A \cdot \operatorname{Cos} 2A}{\operatorname{Sin} A} \][/tex]
Thus, we have the following key steps in solving the equation:
1. Original Equation:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \operatorname{Sin} 4A \cdot \operatorname{Cosec} A \][/tex]
2. Using Trigonometric Identities:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \operatorname{Sin} 4A \cdot \frac{1}{\operatorname{Sin} A} \][/tex]
3. Simplifying the Right Side:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \left(\frac{\operatorname{Sin} 4A}{\operatorname{Sin} A}\right) \][/tex]
4. Applying the Identity for [tex]\( \operatorname{Sin} 4A \)[/tex]:
[tex]\[ \operatorname{Cos} A \cdot \operatorname{Cos} 2A = \frac{1}{4} \cdot 2 \cdot \frac{\operatorname{Sin} 2A \cdot \operatorname{Cos} 2A}{\operatorname{Sin} A} \][/tex]
This brings us to the simplified form of the equation, having manipulated it solely using trigonometric identities and simplification methods.
