Question

If alpha is an acute angle and tan alpha + cot alpha = 2, then the value of sin^3 alpha + cos^3 alpha is (a) 1 (b) b) 2/V2 (c)1/2(d)2^1/2​

Answer 1

Answer:

The correct answer is $\dfrac{1}{\sqrt 2}$

Step-by-step explanation:

Given  $\tan\alpha+\cot\alpha=2$

          $\Rightarrow \dfrac{\sin\alpha}{\cos\alpha}+\dfrac{\cos\alpha}{\sin\alpha}=2\\\\\Rightarrow \dfrac{\sin^2\alpha+\cos^2\alpha}{\sin\alpha\cos\alpha}=2\\\\\Rightarrow \dfrac{1}{\sin\alpha\cos\alpha}=2\\\Rightarrow \sin\alpha\cos\alpha=\dfrac{1}{2}$

Let   $z=\sin^3\alpha+\cos^3\alpha$

          $=(\sin\alpha+\cos\alpha)(1-\dfrac{1}{2})$

          $=\dfrac{\sin\alpha+\cos\alpha}{2}$

 $\Rightarrow z^2=\dfrac{\sin^2\alpha+\cos^2\alpha+2\sin\alpha\cos\alpha}{4}=\dfrac{1+1}{4}=\dfrac{1}{2}\\\Rightarrow z=\dfrac{1}{\sqrt 2}$