Question

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

We observe that multiple of tan function from $1^{\circ }$ to ${89}^{\circ }$ is given in the question and all the angles given in the question lie in the first quadrant.

We know, in first quadrant all the trigonometric ratios give positive value.

We know tan ${89}^{\circ }$= tan (90-1) = cot $1^{\circ }$,

$\Rightarrow$ tan $1^{\circ }$ $\times$ tan ${89}^{\circ }$ = tan $1^{\circ }$ $\times$ cot $1^{\circ }$=1

Similarly,

tan $2^{\circ }$ $\times$ tan ${88}^{\circ }$ = 1

So, $tan1\times tan2\times tan3.............tan87,tan88,tan89$

= $tan1\times tan2\times tan3...................cot3\times cot2\times cot1$

= tan ${45}^{\circ }$ = 1

Answer is 1.