Question
Prove that $\frac{\sin\text{A}-2\sin^3\text{A}}{\big(2\cos^2\text{A}-\cos\text{A}\big)}=\tan\text{A}.$
✓
Answer
$\text{LHS}=\frac{\big(\sin\text{A}-2\sin^2\text{A}\big)}{\Big(2\cos^2\text{A}-\cos\text{A}\big)}$$=\frac{\sin\text{A}\big(1-2\sin^2\text{A}\big)}{\cos\text{A}\big(2\cos^2\text{A}-1\big)}$
$=\tan\text{A}\Bigg\{\frac{\big(\sin^2\text{A}+\cos^2\text{A}-2\sin^2\text{A}\big)}{2\cos^2\text{A}-\sin^2\text{A}-\cos^2\text{A}}\Bigg\}$ $\big[\because\sin^2\text{A}+\cos^2\text{A}=1\big]$
$=\tan\text{A}\Bigg\{\frac{\big(\cos^2\text{A}-\sin^2\text{A}\big)}{\big(\cos^2\text{A}-\sin^2\text{A}\big)}\Bigg\}$
$=\tan\text{A}$
$=\text{RHS}$
$=\tan\text{A}\Bigg\{\frac{\big(\sin^2\text{A}+\cos^2\text{A}-2\sin^2\text{A}\big)}{2\cos^2\text{A}-\sin^2\text{A}-\cos^2\text{A}}\Bigg\}$ $\big[\because\sin^2\text{A}+\cos^2\text{A}=1\big]$
$=\tan\text{A}\Bigg\{\frac{\big(\cos^2\text{A}-\sin^2\text{A}\big)}{\big(\cos^2\text{A}-\sin^2\text{A}\big)}\Bigg\}$
$=\tan\text{A}$
$=\text{RHS}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the lenght of the altitude of an equilateral triangle of side 2a cm.
For what value of x are the points A(-3, 12), B(7, 6) and C(x, 9) colinear?
Show that the cube of a positive integer is of the form $6q + r$, where q is an integer and $r = 0, 1, 2, 3, 4, 5.$
→The area of a parallelogram is $392 m^2$. If its altitude is twice the corresponding base, determine the base and the altitude.
→Solve the following systems of equations graphically:
2x - 3y + 13 = 0
3x - 2y + 12 = 0
From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest $cm^2$.
→A farmer connects a pipe of internal diameter 25cm from a canal into a cylindrical tank in his field, which is 12m in diameter and 2.5m deep. If water flows through the pipe at the rate of 3.6km/hr, then in how much time Will the tank be filled? Also, find the cost of water if the canal department charges at the rate of ₹ 0.07 per $m^3$.
→A wooden article was made by scooping out a hemisphere from each end of a cylinder, as shown in the figure. If the height of the cylinder is 20cm and its base is of diameter 7cm, find the total surface area of the article when it is ready.
Solve: $\frac{\text{ax}}{\text{b}}-\frac{\text{by}}{\text{a}}=\text{a}+\text{b},$ $\text{ax}-\text{by}=\text{2ab}.$
→If $3\cot\theta=2,$ show that $\frac{(4\sin\theta-3\cos\theta)}{(2\sin\theta+6\cos\theta)}=\frac13.$
→