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  4. The product (1+ tan 1°) (1+ tan 2°) (1+ tan 3°) dots (1+ tan 45°) equals

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Q. The product $(1+\tan\,1^{\circ}) (1+\tan\,2^{\circ}) (1+\tan \,3^{\circ})\dots (1+\tan\,45^{\circ})$ equals

KVPYKVPY 2010

Solution:

We have,
$(1+\tan\,1^{\circ}) (1+\tan\,2^{\circ}) (1+\tan\,3^{\circ}) \dots (1+\tan\,45^{\circ})$
We know that,
$(1+\tan\,\theta)(1+\tan(45^{\circ}-\theta))=2$
$\therefore (1+\tan\,1^{\circ}) (1+\tan\,44^{\circ}) (1+\tan\,2^{\circ})$
$(1+\tan\,43^{\circ})\dots (1+\tan\,22^{\circ}) (1+\tan\,23^{\circ})$
$(1+\tan\,45^{\circ})$
$\Rightarrow 2^{22}. 2=2^{23}$