Finding the Angle

Given that $\tan\theta = 1$ and $\theta$ is an acute angle.

The angle $\theta$ for which the tangent is 1 is $45^\circ$. So, $\theta = 45^\circ$.

Calculating Trigonometric Values

For $\theta = 45^\circ$, we find the required trigonometric values:

• $\sin\theta = \sin(45^\circ) = \frac{1}{\sqrt{2}}$
• $\cos\theta = \cos(45^\circ) = \frac{1}{\sqrt{2}}$
• $\text{cosec}\theta = \frac{1}{\sin\theta} = \frac{1}{1/\sqrt{2}} = \sqrt{2}$
• $\text{cosec}^2\theta = (\sqrt{2})^2 = 2$

Evaluating the Expression

The expression is $2 \sin\theta \cos\theta - \text{cosec}^2\theta$. Substitute the calculated values:

$ 2 \sin\theta \cos\theta - \text{cosec}^2\theta = 2 \left( \frac{1}{\sqrt{2}} \right) \left( \frac{1}{\sqrt{2}} \right) - 2 $

Simplify the expression:

$ = 2 \left( \frac{1}{2} \right) - 2 $

$ = 1 - 2 $

$ = -1 $

Thus, the value of the expression is $-1$.