Question

if tan theta equals to root 2 -1 then sin Theta.cos Theta =?​

Answer 1

$\text{Given}$

$tan\theta=\sqrt{2}-1$

$sin\,2\theta={\frac{2\,tan\theta}{1+tan^2\theta}}$

$={\frac{2(\sqrt2-1)}{1+(\sqrt2-1)^2}}$

$={\frac{2(\sqrt2-1)}{1+2+1-2\sqrt2}$

$={\frac{2(\sqrt2-1)}{4-2\sqrt2}$

$\implies\,sin\,2\theta={\frac{2(\sqrt2-1)}{2(2-\sqrt2)}$

$\implies\,2\,sin\theta\,cos\theta={\frac{\sqrt2-1}{2-\sqrt2}}\times{\frac{2+\sqrt2}{2+\sqrt2}}$

$={\frac{2\sqrt2+2-2-\sqrt2}{4-2}}$

$2\,sin\theta\,cos\theta={\frac{\sqrt2}{2}}$

$sin\theta\,cos\theta={\frac{\sqrt2}{4}}$

$\implies\boxed{\bf\,sin\theta\,cos\theta=\frac{1}{2\sqrt2}}$