1+(sinA/tanB)^2/1+(sinA/tanC)^2
1 + (sinA / tanB)^2 / 1 + (sinA / tanC)^2 = (tanB^2 + sinA^2) / (tanC^2 + sinA^2)
To simplify the given expression: 1 + (sinA/tanB)^2 / 1 + (sinA/tanC)^2
First, let's simplify (sinA/tanB)^2 and (sinA/tanC)^2 separately by using trigonometric identities.
Using the identity: tan(X) = sin(X) / cos(X)
Rewriting the given expression:
1 + (sinA/(sinB/cosB))^2 / 1 + (sinA/(sinC/cosC))^2
Next, we simplify the denominators:
sinB/cosB = tanB
sinC/cosC = tanC
Substituting these back into the expression:
1 + (sinA/tanB)^2 / 1 + (sinA/tanC)^2
Now, we have:
1 + (sinA/tanB)^2 / 1 + (sinA/tanC)^2
We can further simplify by finding a common denominator for the fractions in the numerator:
1 + ((sinA)^2 / (tanB)^2) / 1 + ((sinA)^2 / (tanC)^2)
Next, we can combine the fractions in the numerator:
1 + (sinA)^2(tanC)^2 / (tanB)^2(tanC)^2 / 1 + (sinA)^2(tanB)^2 / (tanC)^2(tanB)^2
Since we have a common denominator, we can combine the fractions by adding the numerators together:
1 + (sinA)^2(tanC)^2 + (sinA)^2(tanB)^2 / (tanB)^2(tanC)^2 / 1 + (sinA)^2(tanB)^2 + (sinA)^2(tanC)^2 / (tanC)^2(tanB)^2
Finally, we can simplify further by combining like terms and adding the fractions in the numerator:
1 + (sinA)^2[(tanC)^2 + (tanB)^2] / (tanB)^2(tanC)^2 / 1 + (sinA)^2[(tanB)^2 + (tanC)^2] / (tanC)^2(tanB)^2
This is the simplified form of the given expression.
To simplify the expression 1 + (sinA/tanB)^2 / 1 + (sinA/tanC)^2, we need to apply the order of operations, which is also known as PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, and Addition and Subtraction from left to right).
1 + (sinA/tanB)^2 / 1 + (sinA/tanC)^2
First, let's simplify the numerator and denominator separately:
Numerator:
(sinA/tanB)^2
Since (sinA/tanB)^2 is raised to the power of 2, we can write it as:
(sinA)^2 / (tanB)^2
Since (sinA)^2 and (tanB)^2 are both squared, we can write them as:
sin^2(A) / tan^2(B)
Denominator:
1 + (sinA/tanC)^2
Now, let's simplify the denominator:
(sinA/tanC)^2
Similar to the numerator, we can write it as:
(sinA)^2 / (tanC)^2
So, the expression becomes:
1 + (sin^2(A) / tan^2(B)) / 1 + (sin^2(A) / tan^2(C))
Next, let's simplify the expression further by combining the fractions:
1 + (sin^2(A) * tan^2(C)) / (tan^2(B) * (1 + sin^2(A)))
Since (1 + sin^2(A)) appears in both the numerator and denominator, we can cancel them out:
1 + (tan^2(C) / tan^2(B))
Finally, the simplified expression is:
1 + tan^2(C)/tan^2(B)