What is the simplified value of $$\left(\frac{sin^231^\circ+sin^259^\circ}{sec^235^\circ-cot^255^\circ}+tan29^\circ cot61^\circ-cosec^261^\circ\right)$$?

= $$\left(\frac{sin^231^\circ+sin^259^\circ}{sec^235^\circ-cot^255^\circ}+tan29^\circ cot61^\circ-cosec^261^\circ\right)$$

= $$\left(\frac{\sin^231^{\circ}+\sin^2\left(90-31\right)^{\circ}}{\sec^235^{\circ}-\cot^2\left(90-35\right)^{\circ}}+\tan\left(90-61\right)^{\circ}\cot61^{\circ}-\operatorname{cosec}^261^{\circ}\right)$$

= $$\left(\frac{\sin^2\ 31^{\circ}+\cos^2\ 31^{\circ}}{\sec^2\ 35^{\circ}-\tan^2\ 35^{\circ}}+\cot\ 61^{\circ}\ \cot61^{\circ}-\operatorname{cosec}^261^{\circ}\right)$$

= $$\left(\frac{\sin^2\ 31^{\circ}+\cos^2\ 31^{\circ}}{\sec^2\ 35^{\circ}-\tan^2\ 35^{\circ}}+\cot^2\ 61^{\circ}-\operatorname{cosec}^261^{\circ}\right)$$

We know that $$\sin^2\ \theta\ +\cos^2\ \theta\ \ =\ 1$$    Eq.(i)

$$\sec^2\ \theta\ -\tan^2\ \theta\ \ =\ 1$$    Eq.(ii)

$$\operatorname{cosec}^2\ \theta\ -\cot^2\ \theta\ \ =\ 1$$    Eq.(ii)

From Eq.(i), Eq.(ii) and Eq.(iii).

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

= 0