Question

If θ  is an acute angle and tan θ  + cot θ = 2, then the value of sin³θ + cos³θ is ​

Answer 1

SOLUTION:

Given That:

→ tan θ  + cot θ = 2

→ tan θ  + 1/tan θ = 2

Let us assume that:

→ tan θ = x.

Therefore:

→ x + 1/x = 2

→ (x² + 1)/x = 2

→ x² + 1 = 2x

→ x² - 2x + 1 = 0

→ (x - 1)² = 0

→ x = 1, 1

Therefore:

→ tan θ  = 1, 0° ≤ θ ≤ 90°

But we know that:

→ tan 45° = 1

→ θ = 45°

Now, we have:

= sin³θ + cos³θ

= (1/√2)³ + (1/√2)³

= 2 × 1/(√2)³

= 2 × 1/2√2

= 1/√2

Therefore:

→ sin³θ + cos³θ = 1/√2 (Answer)

LEARN MORE:

1. Relationship between sides and T-Ratios.

• sin(x) = Height/Hypotenuse
• cos(x) = Base/Hypotenuse
• tan(x) = Height/Base
• cot(x) = Base/Height
• sec(x) = Hypotenuse/Base
• cosec(x) = Hypotenuse/Height

2. Square formulae.

• sin²(x) + cos²(x) = 1
• cosec²(x) - cot²(x) = 1
• sec²(x) - tan²(x) = 1

3. Reciprocal Relationship.

• sin(x) = 1/cosec(x)
• cos(x) = 1/sec(x)
• tan(x) = 1/cot(x)

4. Cofunction identities.

• sin(90° - x) = cos(x)
• cos(90° - x) = sin(x)
• cosec(90° - x) = sec(x)
• sec(90° - x) = cosec(x)
• tan(90° - x) = cot(x)
• cot(90° - x) = tan(x)

5. Even odd identities.

• sin(-x) = -sin(x)
• cos(-x) = cos(x)
• tan(-x) = -tan(x)

Answer 2

Answer: 1/√2

Step-by-step explanation: