Prove each of the following iden tities . A. (1-sin theater)(1+sin theater)/sin square theater= cot square theater. B.sin theater/ 1- cos square theater= cosec theater. C.( sec theater- tan theater)( sec theater + tan theater=1 D. (cosec theater - cot theater )( cosec theater + cot theater = 1. E.cos raised to the power of 4 theater - sin raised to the power of 4 theater = 2cos square theater - 1
A.
(1-sinθ)(1+sinθ)/sin²θ = (1-sin²θ)/sin²θ = cos²θ/sin²θ = cot²θ
B.
sinθ/1-cos²θ = sinθ/sin²θ = 1/sinθ = cosecθ
C.
(secθ-tanθ)(secθ+tanθ) = sec²θ - tan²θ = 1/cos²θ - sin²θ/cos²θ = (1 - sin²θ)/cos²θ = cos²θ/cos²θ = 1
D.
(cosecθ-cotθ)(cosecθ+cotθ) = cosec²θ - cot²θ = 1/sin²θ - cos²θ/sin²θ = (1 - cos²θ)/sin²θ = sin²θ/sin²θ = 1
E.
cos⁴θ - sin⁴θ = (cos²θ + sin²θ)(cos²θ - sin²θ) = cos²θ(1 - sin²θ) - sin²θ(1 - cos²θ) = cos²θ - cos⁴θ - sin²θ + sin⁴θ = 2cos²θ - 1 - 2sin²θ = 2(cos²θ - sin²θ) - 1 = 2cos²(θ) - 1
A. To prove the identity (1-sin^2θ)(1+sin^2θ)/sin^2θ = cot^2θ:
Starting with the left side of the equation:
(1-sin^2θ)(1+sin^2θ)/sin^2θ
Simplifying the numerator:
(1-sin^4θ)/sin^2θ
Using the identity 1 - sin^2θ = cos^2θ:
cos^2θ(1-sin^2θ)/sin^2θ
cos^2θ/cos^2θ
This simplifies to 1 and is equal to the right side of the equation, cot^2θ.
Therefore, the identity is proven.
B. To prove the identity sinθ/(1-cos^2θ) = cosecθ:
Starting with the left side of the equation:
sinθ/(1-cos^2θ)
Using the identity 1 - cos^2θ = sin^2θ:
sinθ/sin^2θ
This can be simplified to 1/sinθ, which is equal to cosecθ.
Therefore, the identity is proven.
C. To prove the identity (secθ - tanθ)(secθ + tanθ) = 1:
Start by multiplying the terms using the difference of squares:
(secθ)^2 - (tanθ)^2
Using the identity sec^2θ = 1 + tan^2θ:
1 + tan^2θ - tan^2θ
The tan^2θ terms cancel out, leaving us with:
1
Therefore, the identity is proven.
D. To prove the identity (cosecθ - cotθ)(cosecθ + cotθ) = 1:
Start by multiplying the terms using the difference of squares:
(cosecθ)^2 - (cotθ)^2
Using the identity cosec^2θ = 1 + cot^2θ:
1 + cot^2θ - cot^2θ
The cot^2θ terms cancel out, leaving us with:
1
Therefore, the identity is proven.
E. To prove the identity cos^4θ - sin^4θ = 2cos^2θ - 1:
Start by factoring the left side as a difference of squares:
(cos^2θ + sin^2θ)(cos^2θ - sin^2θ)
Using the Pythagorean identity cos^2θ + sin^2θ = 1:
1(cos^2θ - sin^2θ)
This simplifies to cos^2θ - sin^2θ.
Now, we can use the identity cos^2θ = 1 - sin^2θ:
(1 - sin^2θ) - sin^2θ
1 - 2sin^2θ
Finally, we can rewrite 2sin^2θ as 2(1 - cos^2θ) using the identity sin^2θ = 1 - cos^2θ:
1 - 2(1 - cos^2θ)
1 - 2 + 2cos^2θ
2cos^2θ - 1
Therefore, the left side is equal to the right side, and the identity is proven.