verify the Pythagorean identity. 1+cot^2θ=csc^2θ
Starting with the left side of the identity:
1 + cot^2θ
= (sin^2θ + cos^2θ) / cos^2θ + sin^2θ / cos^2θ
= [(sin^2θ + cos^2θ) + sin^2θ] / cos^2θ
= (2sin^2θ + cos^2θ) / cos^2θ
= (2sin^2θ / cos^2θ) + 1
= 2(csc^2θ - 1) + 1 (using the reciprocal identities)
= 2csc^2θ - 2 + 1
= 2csc^2θ - 1
Which is equal to the right side of the identity, therefore the Pythagorean identity is verified.
what a lot of work!
Start with the most basic of trig identities:
sin^2θ + cos^2θ = 1
and then just divide by sin^2θ to get
1 + cot^2θ = csc^2θ
To verify the Pythagorean identity 1+cot^2θ=csc^2θ, we will use the definitions of cotangent (cot) and cosecant (csc) in terms of sine (sin) and cosine (cos).
Starting with the left-hand side (LHS):
LHS = 1 + cot^2θ
Using the definition of cotangent:
LHS = 1 + (cos^2θ / sin^2θ)
Combining the fractions:
LHS = (sin^2θ + cos^2θ) / sin^2θ
Since sin^2θ + cos^2θ = 1 (which is a fundamental trigonometric identity), we can substitute this value into the equation:
LHS = 1 / sin^2θ
Using the definition of cosecant:
LHS = csc^2θ
Thus, the left-hand side (LHS) is equal to the right-hand side (RHS) of the Pythagorean identity, and we have verified that 1+cot^2θ=csc^2θ.
That's a much simpler approach, and it's a more direct way to arrive at the Pythagorean identity.
Starting with the basic identity sin^2θ + cos^2θ = 1:
Add cot^2θ to both sides:
sin^2θ + cos^2θ + cot^2θ = 1 + cot^2θ
Recall that cot^2θ = cos^2θ / sin^2θ:
sin^2θ + cos^2θ + cos^2θ / sin^2θ = csc^2θ
Combine the terms with a common denominator:
(sin^2θ * sin^2θ + cos^2θ * sin^2θ + cos^2θ) / sin^2θ = csc^2θ
Simplify using the identity sin^2θ + cos^2θ = 1:
(sin^2θ * (1 - cos^2θ) + cos^2θ) / sin^2θ = csc^2θ
Distribute and simplify:
(sin^2θ - sin^2θ * cos^2θ + cos^2θ) / sin^2θ = csc^2θ
Simplify further by factoring out sin^2θ:
[(1 - cos^2θ) + cos^2θ] / sin^2θ = csc^2θ
Simplify and recognize the expression on the left side of the equation:
1 / sin^2θ = csc^2θ
Rearrange to get the Pythagorean identity:
1 + cot^2θ = csc^2θ
This approach is much quicker and requires less algebraic manipulation compared to the previous method.