Question

If tan theta+cot theta=2 prove tan^3theta+cot^3 theta=2

Answer 1

Answer:

tan∅ + cot∅ = 2 ....(1)

to prove tan³∅ + cot³∅ = 2

equation (1)

= tan∅ + 1/tan∅ = 2

→ (tan²∅ + 1)/tan∅ = 2

→ tan²∅ + 1 = 2tan∅

→ tan²∅ - 2tan∅ + 1= 0

→ (tan∅ - 1)² = 0

→ tan∅ - 1 = 0

→ tan∅ = 1 = tan45°

so ∅ = 45°

put this in above question

tan³∅ + cot³∅ = 2

tan³(45) + cot³(45)

= 1 + 1

= 2

hence proved

as tan45 = 1

hope this will help you

Answer 2

Answer:

The correct answer is the proof shown below that $tan^3theta+cot^3 theta=2.$

Step-by-step explanation:

Given: tan(theta) + cot(theta) = 2

Squaring the equation tan(theta) + cot(theta) = 2 we get,

$tan^2(theta) + 2tan(theta)cot(theta) + cot^2(theta) = 4$

Rearranging this equation to solve for the value of $2tan(theta)cot(theta)$:

$2tan(theta)cot(theta) = 4 - (tan^2(theta) + cot^2(theta))$

Formula:

We know that

$tan^2(theta) + cot^2(theta) = 1$

(this is an identity in trigonometry)

Substituting this value into the equation and simplify to get:

$2tan(theta)cot(theta) = 3$

Cubing the equation $tan(theta) + cot(theta) = 2$ ,

$tan^3(theta) + 3tan^2(theta)cot(theta) + 3tan(theta)cot^2(theta) + cot^3(theta) = 8$

Now put the value of $2tan(theta)cot(theta)$ that we calculated earlier into this equation and simplify to get,

$tan^3(theta) + cot^3(theta) + 3(tan(theta)cot(theta))(tan^2(theta) + cot^2(theta)) = 8 - 3(tan(theta)cot(theta))$

Using the identity formula and substituting the value of 2tan(theta)cot(theta) = 3, we get:

$tan^3(theta) + cot^3(theta) = 2$

Therefore, it is proven that $tan^3(theta) + cot^3(theta) = 2$ given that $tan(theta) + cot(theta) = 2$.

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