What is the value of $$\ \frac{tan^{2}25^\circ}{cosec^{2}65^\circ}+\frac{cot^{2}25^\circ}{sec^{2}65^\circ}\ $$+ 2tan$$\ 20^\circ\ tan\ 45^\circ\ tan\ 70^\circ\ $$?

Expression = $$\ \frac{tan^{2}25^\circ}{cosec^{2}65^\circ}+\frac{cot^{2}25^\circ}{sec^{2}65^\circ}\ + 2tan\ 20^\circ\ tan\ 45^\circ\ tan\ 70^\circ\ $$

= $$\ \frac{tan^{2}25^\circ}{cosec^{2}(90^\circ-25^\circ)}+\frac{cot^{2}25^\circ}{sec^{2}(90^\circ-25^\circ)}\ + 2tan\ 20^\circ\ tan\ 45^\circ\ tan\ (90^\circ-20^\circ\ )$$

Using, $$cosec(90^\circ-\theta)=sec\theta$$

= $$\ \frac{tan^{2}25^\circ}{sec^{2}25^\circ}+\frac{cot^{2}25^\circ}{cosec^{2}25^\circ}\ + 2tan\ 20^\circ\ tan\ 45^\circ\ cot\ 20^\circ\ $$

= $$(\frac{sin^225^\circ}{cos^225^\circ}\times cos^225^\circ)+(\frac{cos^225^\circ}{sin^225^\circ}\times sin^225^\circ)+(2tan20^\circ.cot20^\circ.tan45^\circ)$$

$$\because tan\ \theta. cot\ \theta=1$$

= $$(sin^225^\circ+cos^225^\circ)+(2\times1)$$

= $$1+2=3$$

=> Ans - (C)