Q48 / 90 · KVPY 2010 (SB/SX) · Mathematics

The product $\left(1+\tan 1^{\circ}\right)\left(1+\tan 2^{\circ}\right)\left(1+\tan 3^{\circ}\right) \ldots\left(1+\tan 45^{\circ}\right)$ equals-

  • A
    $2^{21}$
  • B
    $2^{22}$
  • Correct answer
    C
    $2^{23}$
  • D
    $2^{25}$

Solution

$(1+\tan \theta)(1+\tan (45-\theta))=(1+\tan \theta)\left(1+\frac{1-\tan \theta}{1+\tan \theta}\right)=2$ $\left(1+\tan 1^{\circ}\right)\left(1+\tan 44^{\circ}\right)=2$ etc so product $=2^{22}\left(1+\tan 45^{\circ}\right)=2^{23}$

Hence, the correct answer is option: 'C'.

More Trigonometric Ratios & Identities Questions