Question

Find the value of $\tan 1^{\circ }\tan 2^{\circ }\tan 3^{\circ }.......\tan {89}^{\circ }$.

Answer 1

Find the value of the given expression.

Given: $\tan 1^{\circ }\tan 2^{\circ }\tan 3^{\circ }.......\tan {89}^{\circ }$.

By solving the given expression we get,

$$\begin{gathered} \tan 1^{\circ }\tan 2^{\circ }\tan 3^{\circ }.......\tan {89}^{\circ }=\left(\tan 1^{\circ }\tan {89}^{\circ }\right)\left(\tan 2^{\circ }\tan {88}^{\circ }\right)......\left(\tan {44}^{\circ }\tan {46}^{\circ }\right)\tan {45}^{\circ } \\ =\left(\tan 1^{\circ }\tan \left({90}^{\circ }-1^{\circ }\right)\right)\left(\tan 2^{\circ }\tan \left({90}^{\circ }-2^{\circ }\right)\right).......\left(\tan {44}^{\circ }\tan \left({90}^{\circ }-{44}^{\circ }\right)\right)\tan {45}^{\circ } \\ =\left(\tan 1^{\circ }\cot 1^{\circ }\right)\left(\tan 2^{\circ }\cot 2^{\circ }\right).......\left(\tan {44}^{\circ }\cot {44}^{\circ }\right)\tan {45}^{\circ }\mathbf{}\mathbf{[}\mathbf{}\mathbf{\because }\mathit{t}\mathit{a}\mathit{n}\mathbf{(}{\mathbf{90}}^{\mathbf{\circ }}\mathbf{-}\mathbf{\theta }\mathbf{)}\mathbf{=}\mathbf{cot}\mathit{\theta }\mathbf{]} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}=\left(\tan 1^{\circ }\times \frac{1}{\tan 1^{\circ }}\right)\left(\tan 2^{\circ }\times \frac{1}{\times \tan 2^{\circ }}\right).......\left(\tan {44}^{\circ }\times \frac{1}{\tan {44}^{\circ }}\right)\times 1\mathbf{[}\mathbf{}\mathbf{\because }\mathbf{cot}\mathbf{=}\frac{\mathbf{1}}{\mathbf{tan\theta }}\mathbf{}\mathbf{and}\mathbf{}\mathbf{tan}{\mathbf{45}}^{\mathbf{\circ }}\mathbf{=}\mathbf{1}\mathbf{]} \\ =\left(1\right)\times \left(1\right)..........\times \left(1\right) \\ =1 \end{gathered}$$

Hence, the value of $\tan 1^{\circ }\tan 2^{\circ }\tan 3^{\circ }.......\tan {89}^{\circ }$is $1$.