csc x - cot x = root of 2 cot x then proof that csc x + cot x = root of 2 csc x
is that √(2 cotx) or (√2)cotx?
(√2)cotx
To prove that csc(x) + cot(x) = √2 csc(x), start with the given equation:
csc(x) - cot(x) = √2 cot(x)
First, let's manipulate the equation using trigonometric identities. We'll focus on the right side: √2 cot(x).
Recall that:
cot(x) = 1/tan(x)
√2 can be rewritten as √2/1
So, we have:
√2 cot(x) = √2 * 1/tan(x)
= √2/tan(x)
Next, let's work on the left side: csc(x) - cot(x).
We can rewrite csc(x) as 1/sin(x). So, the equation becomes:
1/sin(x) - cot(x) = √2/tan(x)
To manipulate the left side further, we need a common denominator. The common denominator for sin(x) and tan(x) is sin(x) * cos(x). So, multiply the numerator and denominator of 1/sin(x) by cos(x) to get:
(cos(x)/sin(x)) - cot(x) = √2/tan(x)
Now, let's express cot(x) in terms of cos(x) and sin(x).
cot(x) = cos(x)/sin(x)
Substituting this back into the equation, we have:
(cos(x)/sin(x)) - (cos(x)/sin(x)) = √2/tan(x)
The denominators are equal, so we can combine the numerators:
(cos(x) - cos(x))/sin(x) = √2/tan(x)
The numerator simplifies to zero, so we have:
0/sin(x) = √2/tan(x)
Any number divided by zero is undefined, so this indicates that the left side is undefined. But since the right side of the equation is defined, there is no solution for this equation.
Therefore, the proof that csc(x) + cot(x) = √2 csc(x) is unsuccessful since the original equation does not hold true.