A^2 b^2a^2 b^2-c^2/c^2 a^2-b^2=tanA/tanB?

Question

Answer

Solution:

Step 1: Simplifying the left-hand side

We start by simplifying the left-hand side of the equation:

A²B²A²B² - C² / C²A² - B²

Expanding A²B²A²B², we get:

(AxAxA)(BxBxB)(AxAxA)(BxBxB) - C² / C²A² - B²

Which simplifies to:

A^6B^6 - C^2 / C^2(A^2 - B^2)

Using the identity A² - B² = (A + B)(A - B), we can simplify further:

A^6B^6 - C^2 / C²(A + B)(A - B)

Now we can simplify the expression by factoring out A²B²:

A²B²(A²B² - C²) / C²(A + B)(A - B)

Step 2: Simplifying the right-hand side

Now we simplify the right-hand side of the equation:

tanA / tanB

Using the identity tanA = sinA / cosA and tanB = sinB / cosB, we get:

sinAcosB / cosAsinB

Which simplifies to:

sinA / sinB * cosB / cosA

Using the identity sin²θ + cos²θ = 1, we can rewrite this as:

(sinA / sinB) * √(1 - sin²B) / √(1 - sin²A)

Step 3: Equating the left-hand side and right-hand side

Now we equate the left-hand side and right-hand side of the equation:

A²B²(A²B² - C²) / C²(A + B)(A - B) = (sinA / sinB) * √(1 - sin²B) / √(1 - sin²A)

Using the identity sin²θ + cos²θ = 1, we can simplify the expression:

A²B²(A²B² - C²) / C²(A + B)(A - B) = (sinA / sinB) * cosB / cosA

Now we can simplify further by cross-multiplying and canceling out common factors:

A²B²(A²B² - C²)cosB = C²(sinA)(A + B)(A - B)cosA

Dividing both sides by A²B²(A²B² - C²)cosAcosB, we get:

(sinA / sinB) = C / √(A²B² - C²)

Now we can use the identity A