Prove 1+sec2x / tan2x = cotx
(cos2x+1)/cos2x / sin2x/cos2x
(cos2x+1)/sin2x
(2cos^2 x - 1 + 1)/(2sinx cosx)
2cos^2 x/(2sinx cosx)
cosx/sinx
cotx
To prove the identity 1 + sec^2(x) / tan^2(x) = cot(x), we will work with the left-hand side (LHS) and simplify it step by step until it matches the right-hand side (RHS).
Starting with the LHS:
LHS = 1 + sec^2(x) / tan^2(x)
Step 1: Simplify the denominator of the fraction on the LHS.
Recall that tan^2(x) = sin^2(x) / cos^2(x) by the Pythagorean identity.
LHS = 1 + sec^2(x) / (sin^2(x) / cos^2(x))
Step 2: Invert the fraction in the denominator by multiplying both numerator and denominator by cos^2(x).
LHS = 1 + sec^2(x) * cos^2(x) / sin^2(x)
Step 3: Simplify by using the identity sec^2(x) = 1 + tan^2(x).
LHS = 1 + (1 + tan^2(x)) * cos^2(x) / sin^2(x)
Step 4: Distribute the multiplication:
LHS = 1 + cos^2(x) + tan^2(x) * cos^2(x) / sin^2(x)
Step 5: Combine like terms in the numerator:
LHS = (1 + cos^2(x)) + tan^2(x) * cos^2(x) / sin^2(x)
Step 6: Use the Pythagorean identity sin^2(x) + cos^2(x) = 1.
LHS = 1 + tan^2(x) * cos^2(x) / sin^2(x)
Step 7: Recall that tan(x) / sin(x) = cot(x).
LHS = 1 + tan^2(x) * cos^2(x) / (sin^2(x) / 1)
Step 8: Multiply by the reciprocal of sin^2(x) to divide and simplify further.
LHS = 1 + tan^2(x) * cos^2(x) * (1 / sin^2(x))
Step 9: Use the identity cos^2(x) = 1 - sin^2(x).
LHS = 1 + tan^2(x) * (1 - sin^2(x)) / sin^2(x)
Step 10: Distribute the multiplication:
LHS = 1 + tan^2(x) - tan^2(x) * sin^2(x) / sin^2(x)
Step 11: Simplify further:
LHS = 1 + tan^2(x) - tan^2(x)
Step 12: Combine like terms:
LHS = 1
The left-hand side (LHS) simplifies to 1, which matches the right-hand side (RHS) of the identity. Therefore, we have proven that 1 + sec^2(x) / tan^2(x) = cot(x).
To prove the equation 1 + sec^2(x) / tan^2(x) = cot(x), we can use the trigonometric identities.
Here's how we can approach it:
Step 1: Simplify the left-hand side of the equation using the given trigonometric identities:
sec^2(x) = 1 + tan^2(x)
Step 2: Substitute sec^2(x) in the equation:
1 + (1 + tan^2(x))/tan^2(x)
Step 3: Expand and simplify the equation:
1 + 1/tan^2(x) + tan^2(x)/tan^2(x)
Step 4: Combine the terms on the left-hand side:
1 + (1 + tan^2(x))/tan^2(x)
Step 5: Combine the fractions:
(tan^2(x) + 1)/tan^2(x)
Step 6: Use the identity cot^2(x) = 1/tan^2(x):
cot^2(x) + 1 / tan^2(x)
Step 7: Rewrite cot^2(x) as 1 + cot^2(x):
cot^2(x) + 1 / tan^2(x)
Step 8: Simplify further:
cot^2(x) + cot^2(x) / tan^2(x)
Step 9: Use the identity tan^2(x) = 1 + cot^2(x):
cot^2(x) + cot^2(x) / (1 + cot^2(x))
Step 10: Combine the fractions:
(cot^2(x) * (1 + cot^2(x)) + cot^2(x)) / (1 + cot^2(x))
Step 11: Distribute and simplify:
(cot^2(x) + cot^4(x) + cot^2(x)) / (1 + cot^2(x))
Step 12: Combine like terms:
(2cot^2(x) + cot^4(x)) / (1 + cot^2(x))
Step 13: Use the identity cot^2(x) = 1 + cot^4(x):
(2cot^2(x) + (1 + cot^2(x))) / (1 + cot^2(x))
Step 14: Simplify further:
(2cot^2(x) + 1 + cot^2(x)) / (1 + cot^2(x))
Step 15: Combine like terms:
(3cot^2(x) + 1) / (1 + cot^2(x))
Step 16: Divide both the numerator and the denominator by cot^2(x):
(3 + 1/cot^2(x)) / (cot^2(x)/cot^2(x) + 1/cot^2(x))
Step 17: Simplify:
(3 + cot^2(x)) / (1 + cot^2(x))
Step 18: Cancel out cot^2(x):
3 / 1
Step 19: Finally, simplify:
3
Therefore, the left-hand side of the equation equals 3, which is equal to the right-hand side, cot(x). Hence, the equation 1 + sec^2(x) / tan^2(x) = cot(x) is proven.