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3.Trigonometrical Ratios, Functions and Identities
normal
The product $\left(1+\tan 1^{\circ}\right)\left(1+\tan 2^{\circ}\right)\left(1+\tan 3^{\circ}\right)$ $. .\left(1+\tan 45^{\circ}\right)$ equals
A
$2^{21}$
B
$2^{22}$
C
$2^{23}$
D
$2^{25}$
(KVPY-2010)
Solution
(c)
We have,
$\begin{array}{l}\left(1+\tan 1^{\circ}\right)\left(1+\tan 2^{\circ}\right)\left(1+\tan 3^{\circ}\right) \\ \text { We know that, } \\ (1+\tan \theta)\left(1+\tan \left(45^{\circ}-\theta\right)\right)=2 \\ \therefore\left(1+\tan 45^{\circ}\right) \\ \left(1+\tan 43^{\circ}\right)\left(1+\tan 44^{\circ}\right)\left(1+\tan 2^{\circ}\right) \\ \left(1+\tan 45^{\circ}\right) \\ \Rightarrow \quad 2^{22} \cdot 2=2^{23} \end{array}$
Standard 11
Mathematics
