If sinθ =cotθ , then the value of cos^2 θ + 2cos^3 θ + cos^4 θ is
(A) 2 (B) 3 (C) 1 (D) 0 (E) none of the above
given: sinθ =cotθ
sinθ = cosθ/sinθ
cosθ = sin^2 θ
cosθ = 1 - cos^2 θ
cos^2 θ = 1 - cosθ
then cos^2 θ + 2cos^3 θ + cos^4 θ
= cos^2 θ(1 + 2cosθ + cos^2 θ)
= (1 - cosθ)(1 + 2cosθ + 1 - cosθ)
= (1 - cosθ)(2 + cosθ)
= 2 - cosθ - cos^2 θ
= 2 - cosθ - (1 - cosθ)
= 1
thank u!
To find the value of cos^2 θ + 2cos^3 θ + cos^4 θ, we'll utilize the given information that sinθ = cotθ.
Let's start by manipulating the given equation. The cotangent of θ is defined as the reciprocal of the tangent of θ, so we can rewrite the given equation as:
sinθ = 1/tanθ
Next, we can replace sinθ with its equivalent value using the relation between sinθ, cosθ, and tanθ in a right triangle:
sinθ = √(1 - cos^2 θ)
Now, let's substitute this expression into our equation:
√(1 - cos^2 θ) = 1/tanθ
Squaring both sides of the equation, we have:
1 - cos^2 θ = 1/tan^2 θ
Now we can express tanθ as the ratio of sine and cosine:
1 - cos^2 θ = 1/(sin^2 θ/cos^2 θ)
Simplifying the right side of the equation, we have:
1 - cos^2 θ = cos^2 θ/sin^2 θ
Now we can cross-multiply to eliminate the denominators:
1 - cos^2 θ = cos^2 θ*(1/sin^2 θ)
1 - cos^2 θ = cos^2 θ*(csc^2 θ)
Now, let's bring all the terms to one side of the equation:
1 - cos^2 θ - cos^2 θ*(csc^2 θ) = 0
Factor out the common term of cos^2 θ:
cos^2 θ*(-1 - csc^2 θ) + 1 = 0
Using the fact that csc^2 θ = 1 + cot^2 θ, substitute the value:
cos^2 θ*(-1 - (1 + cot^2 θ)) + 1 = 0
cos^2 θ*(-2 - cot^2 θ) + 1 = 0
cos^2 θ*(-2 - (1/tan^2 θ)) + 1 = 0
cos^2 θ*(-2 - (1/(sin^2 θ/cos^2 θ))) + 1 = 0
cos^2 θ*(-2 - (cos^2 θ/sin^2 θ)) + 1 = 0
cos^2 θ*(-2sin^2 θ - cos^2 θ) + sin^2 θ = 0
Expanding the expression:
-2sin^2 θ*cos^2 θ - cos^4 θ + sin^2 θ = 0
Rearranging the terms:
sin^2 θ - 2sin^2 θ*cos^2 θ - cos^4 θ = 0
Now, let's substitute the given equation sinθ = cotθ:
cot^2 θ - 2cot^2 θ*cos^2 θ - cos^4 θ = 0
Replacing cotθ with its equivalent value (1/tanθ):
(1/tanθ)^2 - 2(1/tanθ)^2*cos^2 θ - cos^4 θ = 0
Simplifying:
(1/tan^2 θ) - 2(1/tan^2 θ)*cos^2 θ - cos^4 θ = 0
Now we'll substitute tan^2 θ with its equivalent value (1/cos^2 θ):
1/(cos^2 θ) - 2(1/(cos^2 θ))*cos^2 θ - cos^4 θ = 0
1 - 2 - cos^4 θ = 0
Simplifying further:
-1 - cos^4 θ = 0
cos^4 θ = -1
Since the square of a real number cannot be negative, we have a contradiction in our calculation. Hence, there is no real value of θ that satisfies the given equation sinθ = cotθ.
Therefore, the answer is "none of the above" (E).