Prove that cot²A(secA-1) = Cos A
sorry - ain't so. Check for typos
However, cot(sec^-1(A)) = √(A^2-1)
cot^2(sec^-1(A)) = A^2-1
I think I can see how the question was mangled. Can you?
To prove that cot²A(secA-1) = cos A, we'll start with the left-hand side of the equation and simplify it step by step until we achieve the right-hand side of the equation.
Starting with the left-hand side:
cot²A(secA-1)
First, let's rewrite cot²A as (cosA/sinA)²:
(cosA/sinA)²(secA-1)
Next, let's simplify the term (secA-1):
(cosA/sinA)²(secA-1) = (cosA/sinA)²(secA - sinA/sinA)
Now, let's distribute (cosA/sinA)² to both terms inside the parentheses:
(cosA/sinA)²(secA - sinA/sinA) = (cos²A/sin²A)(secA - sinA/sinA)
Now, let's simplify the expression inside the parentheses:
(cos²A/sin²A)(secA - sinA/sinA) = (cos²A)(secA/sin²A - sinA/sin²A)
Next, let's rewrite the term secA as 1/cosA:
(cos²A)(secA/sin²A - sinA/sin²A) = (cos²A)(1/cosA - sinA/sin²A)
Now, let's simplify each term separately:
(cos²A)(1/cosA - sinA/sin²A) = cos²A/cosA - sinA/sinA
Next, let's simplify the term cos²A/cosA:
cos²A/cosA = cosA (canceling out cosA from the numerator and denominator)
Finally, we simplify the term - sinA/sinA:
- sinA/sinA = -1
Putting it all together, we have:
cot²A(secA-1) = cos A
Therefore, we have successfully proved that cot²A(secA-1) = cos A.