Question

If 0 is an acute angle and tan theta + cot theta = 2, then the value of sin^3theta+ cos^3theta is​

Answer 1

Answer:

1/√2

Step-by-step explanation:

Use identity \cot \theta=\dfrac{1}{\tan \theta}cotθ=

tanθ

1

\tan \theta+\dfrac{1}{\tan \theta}=2tanθ+

tanθ

1

=2

Step 2:

Multiply both sides by tanθ

tan²θ + 1 = 2tanθ

Step 3:

Transpose 2tanθ on LHS and use identity (a - b)² = (a² + b² - 2ab) and solve for tanθ

tan²θ + 1 - 2tanθ = 0

(tanθ - 1)² = 0

tanθ - 1 =0

tanθ = 1

Step 4:

Using tan45° =1 as θ is acute angles Hence

θ = 45°

Step 5:

Using sin45° =1/√2 and cos45° =1/√2 evaluate sin³θ + cos³θ

\left(\dfrac{1}{ \sqrt{2} } \right)^3+\left(\dfrac{1}{ \sqrt{2} } \right)^3(

2

1

)

3

+(

2

1

)

3

=\dfrac{1}{ 2\sqrt{2} }+\dfrac{1}{ 2\sqrt{2} }=

2

2

1

+

2

2

1

=\dfrac{2}{ 2\sqrt{2} }=

2

2

2

=\dfrac{1}{ \sqrt{2} }=

2

1

Answer 2

Answer:

$\frac{1}{ \sqrt{2} }$