Answer:
secA = ±√17/4 cannot be proved using the given expression since it leads to an invalid result.
Explanation:
Start by simplifying the given expression and then proceed to prove the statement.
Given expression: cosA/tanA × sinA = 16
1- First, recall the definitions of trigonometric functions:
tanA = sinA / cosA
secA = 1 / cosA
2- Substitute tanA in terms of sinA and cosA in the given expression:
cosA / (sinA / cosA) × sinA = 16
Now, simplify the expression by canceling out the common factors:
cosA × cosA × sinA / sinA = 16
cos^2A = 16
Now, take the square root of both sides:
cosA = ±√16
cosA = ±4
Next, we know the identity: cos^2A + sin^2A = 1
Substitute the value of cosA from above:
(±4)^2 + sin^2A = 1
16 + sin^2A = 1
Solve for sin^2A:
sin^2A = 1 - 16
sin^2A = -15
Now, we must determine if this is possible. However, sine squared cannot be negative in a real-valued trigonometric context since it is always non-negative (between 0 and 1). This means that there is no real solution for A that satisfies the given expression.
Therefore, the statement that secA = ±√17/4 cannot be proved using the given expression since it leads to an invalid result.