Prove the following trigonometric identities
(i)(1+cotx)^2 +(1-cotx)^2=2cosec^2x
(ii)cos(270degrees+theta)/cos(360degrees -theta)
(1+cotx)^2 = 1 + 2cotx + cot^2x
(1-cotx)^2 = 1 - 2cotx + cot^2x
add 'em up to get
2 + 2cot^2x
= 2(1+cot^2x)
= 2csc^2x
(i) To prove the identity (1+cotx)^2 +(1-cotx)^2=2cosec^2x, we'll start by expanding both sides of the equation and simplifying.
Expanding the left-hand side:
(1+cotx)^2 +(1-cotx)^2
= (1+cotx)(1+cotx) +(1-cotx)(1-cotx)
= 1+cotx+cotx+cot^2(x) +1-cotx-cotx+cot^2(x)
= 2 + 2cot^2(x)
Using the reciprocal identity, cot^2(x) = 1 + cosec^2(x), we can substitute this back into the equation:
= 2 + 2(1+cosec^2(x))
= 2 + 2 + 2cosec^2(x)
= 4 + 2cosec^2(x)
Now, let's simplify the right-hand side of the equation:
2cosec^2(x)
Comparing the left-hand side and the right-hand side, we can see that they are equal:
4 + 2cosec^2(x) = 2cosec^2(x)
Thus, we have proved the identity (1+cotx)^2 +(1-cotx)^2 = 2cosec^2x.
(ii) To simplify the expression cos(270 degrees + theta)/cos(360 degrees - theta), we'll use the periodicity and cofunction identities of trigonometric functions.
First, let's simplify cos(270 degrees + theta):
cos(270 degrees + theta) = cos(270 degrees)cos(theta) - sin(270 degrees)sin(theta)
Since cos(270 degrees) = 0 and sin(270 degrees) = -1, the expression becomes:
0*cos(theta) - (-1)*sin(theta) = sin(theta)
Next, let's simplify cos(360 degrees - theta):
cos(360 degrees - theta) = cos(360 degrees)cos(theta) + sin(360 degrees)sin(theta)
Since cos(360 degrees) = 1 and sin(360 degrees) = 0, the expression becomes:
1*cos(theta) + 0*sin(theta) = cos(theta)
Now, we can substitute these simplified expressions back into the original expression:
sin(theta)/cos(theta)
Using the definition of tangent (tan(theta) = sin(theta) / cos(theta)), we can rewrite this as:
tan(theta)
Therefore, cos(270 degrees + theta)/cos(360 degrees - theta) simplifies to tan(theta).
To prove the trigonometric identities, let's break down each part step by step:
(i) (1+cotx)^2 + (1-cotx)^2 = 2cosec^2x
Step 1: Expand both sides of the equation using the formula (a+b)^2 = a^2 + 2ab + b^2.
On the left side:
(1+cotx)^2 = 1^2 + 2(1)(cotx) + cotx^2
= 1 + 2cotx + cotx^2
On the right side:
2cosec^2x = 2 * (1/sin^2x) = 2/sin^2x
So we have:
1 + 2cotx + cotx^2 + 1 - 2cotx + cotx^2 = 2/sin^2x
Simplifying further:
2cotx^2 + 2 = 2/sin^2x
Dividing both sides by 2:
cotx^2 + 1 = 1/sin^2x
Using the Pythagorean Identity (1 + cot^2x = csc^2x), we can rewrite the left side as:
csc^2x = 1/sin^2x
Hence, the left side of the equation becomes equal to the right side, which proves the identity:
(1+cotx)^2 + (1-cotx)^2 = 2cosec^2x
(ii) cos(270°+θ)/cos(360°-θ)
Step 1: Simplify the angles in terms of their trigonometric equivalents.
270° can be written as 3π/2 radians, while 360° can be expressed as 2π radians.
So, the expression becomes:
cos(3π/2 + θ) / cos(2π - θ)
Step 2: Use the cosine addition formula to simplify the numerator:
cos(3π/2 + θ) = cos(3π/2) * cos(θ) - sin(3π/2) * sin(θ)
= 0 * cos(θ) - (-1) * sin(θ)
= sin(θ)
Step 3: Simplify the denominator:
cos(2π - θ) = cos(2π) * cos(θ) + sin(2π) * sin(θ)
= 1 * cos(θ) + 0 * sin(θ)
= cos(θ)
Combining the numerator and denominator, we get:
sin(θ) / cos(θ) = tan(θ)
Therefore, the identity cos(270°+θ) / cos(360°-θ) = tan(θ) holds true.