Question

If cosθ =tan θ , then the value of 2sinθ +sin^2θ +sin^3θ +sin^4θ is (A) 2 (B)3 (C) 1 (D) 0 (E) none of the above

Answer 1

if cosθ = tanθ then

sinθ = cos^2θ
sinθ = 1 - sin^2θ
sin^2θ = 1 - sinθ
so,
2sinθ +sin^2θ +sin^3θ +sin^4θ
= 2 - 2sin^2θ + 1 - sinθ + sinθ(1 - sinθ) + (1-sinθ)^2
= 3 - 2sinθ + 1 - 2sin^2θ
= 3 - 2sinθ + sinθ - 1 + sinθ
= 2

Answer 2

sin θ / cos θ = cos θ

so
sin θ = cos^2 θ
so
first two terms
2 cos^2 + sin^2 = cos^2 + 1
last two terms
sin^2 (sin + sin^2) = sin^2 (sin + 1 - cos^2) but cos^2 = sin
so sin^2 (sin + 1 - sin) = sin^2
so the sum is
cos^2 + 1 + sin^2 = 1+1 = 2

Answer 3

cute problem :)

Answer 4

THANK YOU!!!!

Answer 5

[AI-generated answer — Explain Bot]

To find the value of 2sinθ + sin^2θ + sin^3θ + sin^4θ, we need to first use the given information that cosθ = tanθ.

Using the definitions of cosine and tangent, we can rewrite the equation as:

cosθ = sinθ / cosθ

Cross-multiplying gives us:

cos^2θ = sinθ

Since cosine squared is equal to 1 - sin squared (from the Pythagorean identity), we can substitute that in:

1 - sin^2θ = sinθ

Rearranging the equation, we get:

sin^2θ + sinθ - 1 = 0

Now, we have a quadratic equation in terms of sinθ. We can solve this equation using factoring or the quadratic formula.

For simplicity, let's substitute a = sinθ:

a^2 + a - 1 = 0

Factoring won't work in this case, so we will use the quadratic formula:

a = (-b ± √(b^2 - 4ac)) / 2a

Plugging in the values:

a = (-(1) ± √(1^2 - 4(1)(-1))) / 2(1)

Simplifying further:

a = (-1 ± √(1 + 4)) / 2

a = (-1 ± √5) / 2

We have two possible solutions for a (sinθ):

a1 = (-1 + √5) / 2
a2 = (-1 - √5) / 2

Now, we can find the value of 2sinθ + sin^2θ + sin^3θ + sin^4θ by substituting each value of a back into the expression.

Let's start with a1:

2sinθ + sin^2θ + sin^3θ + sin^4θ
= 2(a1) + (a1)^2 + (a1)^3 + (a1)^4
= 2((-1 + √5) / 2) + ((-1 + √5) / 2)^2 + ((-1 + √5) / 2)^3 + ((-1 + √5) / 2)^4

Similarly, we can calculate the value for a2.

After calculating the values for both a1 and a2, we can check which option (A, B, C, D) matches the final result.