Derive the identity 1 + cot ^2 theta = csc^2 theta by dividing x^2 + y^2 = r^2 by y^2
This must be in reference to a right-angled triangle in standard position with sides x, y, and r
so that sinØ = y/r, cosØ = x/r and tanØ = y/x
You should also know the three reciprocal identities.
x^2 + y^2 = r^2
divide each term by y^2
x^2/y^2 + 1 = r^2/y^2
or
cot^2 Ø + 1 = csc^2 Ø
Well, that's a great way to ruin a perfectly good circle. Dividing the equation x^2 + y^2 = r^2 by y^2 is like separating a couple who is meant to be together - it's just not right!
But since you insist, let's go ahead and do this mathematical heartbreak. Dividing x^2 + y^2 = r^2 by y^2 gives us:
x^2 / y^2 + y^2 / y^2 = r^2 / y^2
Now, let's simplify the left-hand side:
(x / y)^2 + 1 = (r / y)^2
Ah, we're making progress! Now, let's rewrite x / y as cot(theta) and r / y as csc(theta):
cot^2(theta) + 1 = csc^2(theta)
And voila! We have achieved the heart-wrenching identity 1 + cot^2(theta) = csc^2(theta). It's just like a math breakup - it might make you feel sad, but you know it's necessary for the sake of mathematical clarity.
To derive the identity 1 + cot^2(theta) = csc^2(theta) by dividing x^2 + y^2 = r^2 by y^2, follow these steps:
Step 1: Start with the equation x^2 + y^2 = r^2, where (x, y) represents a point on the unit circle.
Step 2: Divide both sides of the equation by y^2 to isolate x^2/y^2:
x^2/y^2 + y^2/y^2 = r^2/y^2
Step 3: Simplify both sides:
x^2/y^2 + 1 = r^2/y^2
Step 4: Recognize that x/y represents the cotangent (cot) of theta and r/y represents the cosecant (csc) of theta, and rewrite the equation using trigonometric functions:
cot^2(theta) + 1 = csc^2(theta)
Step 5: Rearrange the equation to match the desired identity:
1 + cot^2(theta) = csc^2(theta)
Therefore, by dividing x^2 + y^2 = r^2 by y^2, we derive the identity 1 + cot^2(theta) = csc^2(theta).
To derive the identity 1 + cot^2(theta) = csc^2(theta) by dividing x^2 + y^2 = r^2 by y^2, follow these steps:
Step 1: Start with the equation of a circle in Cartesian coordinates, x^2 + y^2 = r^2.
Step 2: Divide both sides of the equation by y^2 to isolate y^2 on the left-hand side:
x^2/y^2 + y^2/y^2 = r^2/y^2
Now, x^2/y^2 simplifies to (x/y)^2, and y^2/y^2 simplifies to 1:
(x/y)^2 + 1 = r^2/y^2
Step 3: Convert x/y to cot(theta) and r/y to csc(theta) using trigonometric identities. Recall that cot(theta) = x/y and csc(theta) = r/y:
cot^2(theta) + 1 = csc^2(theta)
By substituting cot^2(theta) and csc^2(theta) back into the equation, we obtain the desired identity:
1 + cot^2(theta) = csc^2(theta)
Therefore, by dividing x^2 + y^2 = r^2 by y^2, we can derive the identity 1 + cot^2(theta) = csc^2(theta).