cos2 theta - sin2 theta = 1/2 find tan2 theta
You asked:
Given \(\cos\left( 2 \right) \cdot θ - \sin\left( 2 \right) \cdot θ = \frac{1}{2}\) evaluate the expression \(\tan\left( 2 \right) \cdot θ\).
MathBot Answer:
\[θ \tan{\left(2 \right)} = \frac{\sqrt{2} \tan{\left(2 \right)}}{4 \cos{\left(\frac{\pi}{4} + 2 \right)}}\]
\[\begin{aligned}θ \left(- \sin{\left(2 \right)} + \cos{\left(2 \right)}\right) &= \frac{1}{2}\\θ &= \frac{1}{2 \left(- \sin{\left(2 \right)} + \cos{\left(2 \right)}\right)}\end{aligned}\]
By substituting this value into the expression, we find that: $$\begin{aligned}θ \tan{\left(2 \right)} &= \frac{\sqrt{2}}{4 \cos{\left(\frac{\pi}{4} + 2 \right)}} \tan{\left(2 \right)}\\&= \frac{\sqrt{2} \tan{\left(2 \right)}}{4 \cos{\left(\frac{\pi}{4} + 2 \right)}}\end{aligned}$$