MCQ
The value of $\tan1^\circ\tan2^\circ\tan3^\circ.....\tan89^\circ$ is :
- ✓$1$
- B$-1$
- C$0$
- DNone of these
✓
Answer
Correct option: A.
$1$
Here we have to find : $\tan1^\circ\tan2^\circ\tan3^\circ.....\tan89^\circ$
$\tan1^\circ\tan2^\circ\tan3^\circ.....\tan89^\circ$
$=\tan(90^\circ-89^\circ)\tan(90^\circ-88^\circ)\tan(90^\circ-87^\circ) ...\tan87^\circ\tan88^\circ\tan89^\circ$
$=\cot89^\circ\cot88^\circ\cot87^\circ...\tan87^\circ\tan88^\circ\tan89^\circ$
$=(\cot89^\circ-\tan89^\circ)(\cot88^\circ\tan88^\circ)$
$\ \ \ \ (\cot87^\circ\tan87^\circ)...(\cot44^\circ\tan44^\circ)\tan45^\circ$
$=1\times1\times1...1\times1\ [\text{since}\cot\theta\tan\theta=1]$
$=1$
Hence the correct option is $(a)$
$\tan1^\circ\tan2^\circ\tan3^\circ.....\tan89^\circ$
$=\tan(90^\circ-89^\circ)\tan(90^\circ-88^\circ)\tan(90^\circ-87^\circ) ...\tan87^\circ\tan88^\circ\tan89^\circ$
$=\cot89^\circ\cot88^\circ\cot87^\circ...\tan87^\circ\tan88^\circ\tan89^\circ$
$=(\cot89^\circ-\tan89^\circ)(\cot88^\circ\tan88^\circ)$
$\ \ \ \ (\cot87^\circ\tan87^\circ)...(\cot44^\circ\tan44^\circ)\tan45^\circ$
$=1\times1\times1...1\times1\ [\text{since}\cot\theta\tan\theta=1]$
$=1$
Hence the correct option is $(a)$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If $\text{ABC}$ is an isosceles triangle and $D$ is a point on $BC$ such that $\text{AD}\perp\text{BC},$ then:
→In figure, if PA and PB are tangents to the circle with centre O such that $\angle\text{APB}=50^\circ,$ then $\angle\text{OAB}$ is equal to:
→The $\text{HCF}$ of $135$ and $225$ is :
→Choose the correct answer from the given four options: A metallic spherical shell of internal and external diameters $4\ cm$ and $8\ cm$, respectively is melted and recast into the form a cone of base diameter $8\ cm$. The height of the cone is:
→In $\triangle\text{DEF}$ and $\triangle\text{PQR},$ it is given that $\angle\text{D}=\angle\text{Q}$ and $\angle\text{R}=\angle\text{E},$ then which of the following is not true ?
→If $\triangle A B C \sim \triangle P Q R$ such that 3AB = 2PQ and BC=10 cm, then length QR is equal to
→The radii of two concentric circles are $19 cm$ and $16 cm$ respectively. The area of the ring enclosed by these circles is
→The $7^{th}$ term of an $AP$ is $-1$ its $16^{th}$ term is $17$. The nth term of the $AP$ is:
→Which of the following numbers cannot be the probability of an event?
The distance of the point $(5,-4)$ from $x$-axis is
→