Download App

Trigonometric Functions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsTrigonometric Functions5 Marks
Question
Prove that:
$\tan20^\circ\tan40^\circ\tan60^\circ\tan80^\circ=3$
✓

Answer

$\text{LHS}=\tan20^\circ\tan40^\circ\tan60^\circ\tan80^\circ$
$=\ (\tan20^\circ\tan40^\circ\tan80^\circ)\sqrt3$$[\because\ \tan60^\circ=\sqrt3]$
$=\ \Big(\frac{\sin20^\circ\sin40^\circ\sin80^\circ}{\cos20^\circ\cos40^\circ\cos80^\circ}\Big)\sqrt3$
$=\ \frac{(2\sin20^\circ\sin40^\circ)\sin80^\circ\times\sqrt3}{(2\cos20^\circ\cos40^\circ)\cos80^\circ}$
Applying
$\Rightarrow\ 2\sin\text{A}\sin\text{B}-\cos(\text{A}-\text{B})-\cos(\text{A+B})$
$2\cos\text{A}\cos\text{B}-\cos(\text{A+B})+\cos(\text{A}-\text{B})$
$=\ \frac{(\cos(40^\circ-20^\circ)-\cos(40^\circ+20^\circ))\sin80^\circ\times\sqrt3}{(\cos(20^\circ+40^\circ)+\cos(40^\circ-20^\circ))\cos80^\circ}$
$=\ \frac{(\cos20^\circ-\cos60^\circ)\sin80^\circ\times\sqrt3}{(\cos60^\circ+\cos20^\circ)\cos80^\circ}$
$=\ \frac{\Big(\cos20^\circ-\frac{1}{2}\Big)\sin80^\circ\times\sqrt3}{\Big(\frac{1}{2}+\cos20^\circ\Big)\cos80^\circ}$
$=\ \frac{(2\sin80^\circ\cos20^\circ-\sin80^\circ)\sqrt3}{\cos80^\circ+2\cos20^\circ\cos80^\circ}$
$\Rightarrow\ 2\sin\text{A}\cos\text{B}-\sin(\text{A+B})+\sin(\text{A}-\text{B})$
$=\ \frac{(\sin(80^\circ+20^\circ)+\sin(80^\circ-20^\circ)-\sin80^\circ)\sqrt3}{\cos80^\circ+(\cos(20^\circ+80^\circ)+\cos(80^\circ-20^\circ))}$
$=\ \frac{(\sin100^\circ+\sin60^\circ-\sin80^\circ)\sqrt3}{\cos80^\circ+\cos(180^\circ-80^\circ)+\cos60^\circ}$
$=\ \frac{\Big(\sin(180^\circ+\frac{\sqrt3}{2}-\sin80^\circ)\Big)\sqrt3}{\cos80^\circ-\cos80^\circ+\cos60^\circ}$
$=\ \frac{\frac{3}{2}}{\frac{1}{2}}=3=\text{RHS}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Trigonometric Functions→Trigonometric Functions — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Prove the following statement by principle of mathematical induction:
$2^{3n} - 1$ is divisible by $7,$ for all natural numbers n.
→
Find the equation of the diagonals of the square formed by the lines $x = 0, y = 0, x = 1$ and $y = 1.$
→
The mean and standard deviation of a group of 100 observations were found to be 20 and 3 respectively. Later on it was found that three observation were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observation were omitted.
→
A man arranges to pay off a debt of ₹ 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid, find the value of the first instalment.
→
If $\cos\alpha+\cos\beta=0=\sin\alpha+\sin\beta,$ then prove that $\cos2\alpha+\cos2\beta=-2\cos(\alpha+\beta).$
$\big[$Hint: $(\cos\alpha+\cos\beta)^2-(\sin\alpha+\sin\beta)^2=0\big]$
→
A circle of radius $4$ units touches the coordinate axes in the first quadrant. Find the equations of its images with respect to the line mirrors $x = 0$ and $y = 0.$
→
$\sin^3\text{x}+\sin^3\big(\frac{2\pi}{3}+\text{x}\big)+\sin^3\big(\frac{4\pi}{3}+\text{x}\big)=-\frac{3}{4}\sin3\text{x}$
→
If $\text{f(x)}=\log_\text{e}(1-\text{x})$ and $\text{g(x)}=[\text{x}],$ then determine the following functions:
$\Big(\frac{\text{f}}{\text{g}}\Big)\Big(\frac{1}{2}\Big)$
→
If n is a positive integer, prove that $3^{3\text{n}}-26\text{n}-1$ is divisible by 676.
→
Find the eccentricity, coordinates of foci, length of the latus-rectum of the following ellipes:
$4\text{x}^2+9\text{y}^2=1$
→