Derive the identity cot² A+1= csc² A
To derive the identity cot² A + 1 = csc² A, we can start by expressing cot² A and csc² A in terms of sin A and cos A.
Recall that:
- cot A = cos A / sin A
- csc A = 1 / sin A
Now, let's rewrite cot² A and csc² A using these trigonometric identities.
cot² A = (cos A / sin A)² = cos² A / sin² A
csc² A = (1 / sin A)² = 1 / (sin A)²
Next, let's work with the right-hand side of the equation, csc² A, and see if we can manipulate it to match the left-hand side.
csc² A = 1 / (sin A)²
Now, we can use the Pythagorean identity, which states that sin² A + cos² A = 1.
Rearranging this equation, we have:
sin² A = 1 - cos² A
Substituting this into the expression for csc² A, we get:
csc² A = 1 / (1 - cos² A)
Now, we can see that the right-hand side expression, 1 / (1 - cos² A), is equal to the left-hand side expression, cot² A + 1.
Therefore, cot² A + 1 = csc² A is derived.
To use this identity in practice, you can replace cot² A + 1 with csc² A in any trigonometric equation or problem you are solving.
well, it depends on if you already know that
sin^2 + cos^2 = 1
if you do, then it is easy
divide both sides by sin^2
1 +cos^2/sin^2 = 1/sin^2
that is it