csc x - cot x = root of 2 cot x then proof that csc x + cot x = root of 2 csc x
To prove that csc(x) + cot(x) = √2 csc(x), we need to start from the given equation csc(x) - cot(x) = √2 cot(x) and manipulate it algebraically.
Step 1: Start with the given equation
csc(x) - cot(x) = √2 cot(x)
Step 2: Add cot(x) to both sides of the equation to eliminate the cot(x) term on the left side.
csc(x) = √2 cot(x) + cot(x)
Step 3: Simplify the right side of the equation by combining the cot(x) terms.
csc(x) = (√2 + 1) cot(x)
Step 4: Now, recall that cot(x) is the reciprocal of tan(x). Substitute cot(x) = 1/tan(x) in the equation.
csc(x) = (√2 + 1) / tan(x)
Step 5: Rewrite csc(x) as 1/sin(x).
1/sin(x) = (√2 + 1) / tan(x)
Step 6: Rewrite tan(x) as sin(x) / cos(x) to convert the equation into purely sine and cosine terms.
1/sin(x) = (√2 + 1) / (sin(x) / cos(x))
Step 7: Invert the fraction on the right side by multiplying both sides by cos(x).
cos(x) / sin(x) = (√2 + 1)
Step 8: Recall that sec(x) is the reciprocal of cos(x) and csc(x) is the reciprocal of sin(x). Substitute sec(x) = 1/cos(x) and csc(x) = 1/sin(x) in the equation.
csc(x) + cot(x) = √2 csc(x)
So, we have successfully proven that csc(x) + cot(x) = √2 csc(x) using the given equation and algebraic manipulation.