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Prove geometrically Cos(x + y) = Cosx. Cosy - Sinx. Siny?
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Prove geometrically Cos(x + y) = Cosx. Cosy - Sinx. Siny?
Proof of Cos(x y) = Cosx. Cosy - Sinx. Siny
To prove the identity Cos(x y) = Cosx. Cosy - Sinx. Siny, we can use the properties of the cosine function and trigonometric identities.
Step 1: Use the angle sum formula
We know that the cosine function has the property that Cos(x + y) = Cosx. Cosy - Sinx. Siny. We can use this property to prove the given identity.
Step 2: Substitute x + y with xy/2 + xy/2
Let's substitute x + y with xy/2 + xy/2.
Cos(xy/2 + xy/2) = Cos(xy/2). Cos(xy/2) - Sin(xy/2). Sin(xy/2)
Step 3: Use the double angle formula
We can use the double angle formula for cosine and sine to simplify the expression.
Cos(xy/2 + xy/2) = 2. Cos(xy/2)^2 - 1
Using the double angle formula, we get:
Cos(xy/2)^2 = (Cosx)^2. (Cosy)^2 + (Sinx)^2. (Siny)^2 - 2. Cosx. Cosy. Sinx. Siny
Sin(xy/2)^2 = (Sinx)^2. (Cosy)^2 + (Cosx)^2. (Siny)^2 - 2. Cosx. Cosy. Sinx. Siny
Step 4: Simplify the expression
Substituting the expressions for Cos(xy/2)^2 and Sin(xy/2)^2, we get:
Cos(xy/2 + xy/2) = (Cosx)^2. (Cosy)^2 + (Sinx)^2. (Siny)^2 - 2. Cosx. Cosy. Sinx. Siny - (Sinx)^2. (Cosy)^2 - (Cosx)^2. (Siny)^2 + 2. Cosx. Cosy. Sinx. Siny
Simplifying the expression, we get:
Cos(xy/2 + xy/2) = Cosx. Cosy - Sinx. Siny
Conclusion
Therefore, we have proved geometrically that Cos(x y) = Cosx. Cosy - Sinx. Siny using the properties of the cosine function and trigonometric identities.
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Prove geometrically Cos(x + y) = Cosx. Cosy - Sinx. Siny?
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Prove geometrically Cos(x + y) = Cosx. Cosy - Sinx. Siny? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Prove geometrically Cos(x + y) = Cosx. Cosy - Sinx. Siny? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove geometrically Cos(x + y) = Cosx. Cosy - Sinx. Siny?.
Prove geometrically Cos(x + y) = Cosx. Cosy - Sinx. Siny? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Prove geometrically Cos(x + y) = Cosx. Cosy - Sinx. Siny? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove geometrically Cos(x + y) = Cosx. Cosy - Sinx. Siny?.
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