csc^2x
1+cot^2x
Recall the Pythagorean Identity:
– sin^2x + cos^2x = 1
Dividing both sides by cos^2x, we get:
– sin^2x/cos^2x + cos^2x/cos^2x = 1/cos^2x
Using trig identities:
– tan^2x + 1 = sec^2x
Substituting in sin^2x/cos^2x for tan^2x:
– sin^2x/cos^2x + 1 = sec^2x
Subtracting 1 from both sides:
– sin^2x/cos^2x = sec^2x – 1
Now we can substitute in the reciprocal identity for secant:
– sin^2x/cos^2x = (1/cos^2x) – 1
Combining like terms:
– sin^2x/cos^2x = (1-cos^2x)/cos^2x
Using the Pythagorean Identity again:
– sin^2x/cos^2x = sin^2x/(1-sin^2x)
Therefore:
– csc^2x = (1-sin^2x)/sin^2x
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