Question
If $\cot\theta=\frac{1}{\sqrt{3}},$ write the value of $\frac{1-\cos^2\theta}{2-\sin^2\theta}.$
✓
Answer
We have, $\cot\theta=\frac{1}{\sqrt{3}}$ In $\triangle\text{ABC},$
$\text{AC}^2=\text{AB}^2+\text{BC}^2$ $\Rightarrow\text{AC}^2=(\sqrt{3})^2+(1)^2$ $\Rightarrow\text{AC}^2=3+1$ $\Rightarrow\text{AC}^2=4$ $\Rightarrow\text{AC}^2=4$ $\therefore\cos\theta=\frac{\text{BC}}{\text{AC}}=\frac{1}{2}$ and $\sin\theta=\frac{\text{AB}}{\text{AC}}=\frac{\sqrt{3}}{2}$ Now, $\frac{1-\cos^2\theta}{2-\sin^2\theta}=\frac{1-\Big(\frac{1}{2}\Big)^2}{2-\Big(\frac{\sqrt{3}}{2}\Big)^2}$ $=\frac{1-\frac{1}{4}}{2-\frac{3}{4}}$ $=\frac{3}{5}$
$\text{AC}^2=\text{AB}^2+\text{BC}^2$ $\Rightarrow\text{AC}^2=(\sqrt{3})^2+(1)^2$ $\Rightarrow\text{AC}^2=3+1$ $\Rightarrow\text{AC}^2=4$ $\Rightarrow\text{AC}^2=4$ $\therefore\cos\theta=\frac{\text{BC}}{\text{AC}}=\frac{1}{2}$ and $\sin\theta=\frac{\text{AB}}{\text{AC}}=\frac{\sqrt{3}}{2}$ Now, $\frac{1-\cos^2\theta}{2-\sin^2\theta}=\frac{1-\Big(\frac{1}{2}\Big)^2}{2-\Big(\frac{\sqrt{3}}{2}\Big)^2}$ $=\frac{1-\frac{1}{4}}{2-\frac{3}{4}}$ $=\frac{3}{5}$
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 $p, q$ are real and $p \neq q$, then show that the show that the roots of the equation $(p-q) x^2+5(p+q) x-2(p-q)=0$ are real and unequal.
→The third term of an $A.P.$ is $7$ and the seventh term exceeds three times the third term by $2$. Find the first term, the common difference and the sum of first $20$ terms.
→A 13-m-long ladder reaches a window of a building 12m above the ground. Determine the distance of the foot of the ladder from the building.
A field is in the form of a circle. A fence is to be erected around the field. The cost fencing would be Rs. 2640 at the rate of Rs. 12 per metre. The, the field is to be thoroughly ploughed at the cost of Re. 0.50 per $m ^2$. What is the amount required to plough the field?
$\left[\right.$ Take $\left.\pi=\frac{22}{7}\right]$.
→$\left[\right.$ Take $\left.\pi=\frac{22}{7}\right]$.
A lending library has a fixed charge for the first three days and additional charge for each day thereafter. Saritha paid Rs. 27 for a book kept for seven days, while Susy paid Rs. 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.
In an isosceles triangle $ABC$, if $AB = AC = 13\ cm$ and the altitude from $A$ on $BC$ is $5cm$, find $BC.$
→If the point $C(-1,2)$ divides internally the line segment joining the points $A(2,5)$ and $B(x, y)$ in the ratio $3: 4$, find the value of $x^2+y^2$.
→Prove the following trigonometric identities.
If $\cos\text{A}+\cos^2\text{A}=1,$ prove that $\sin^2\text{A}+\sin^4\text{A}=1.$
→If $\cos\text{A}+\cos^2\text{A}=1,$ prove that $\sin^2\text{A}+\sin^4\text{A}=1.$
Solve the following quadratic equation:$\frac{2}{\text{x}^2}-\frac{5}{\text{x}}+2=0$
→In the figure, PQ is a chord of a circle and PT is the tangent at P such that $\angle\text{QPT}=60^{\circ}.$ Then, find $\angle\text{PRQ}$.
→