Download App

Inverse Trigonometric Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsInverse Trigonometric Functions5 Marks
Question
For any a, b, x, y > 0, prove that:
$\frac{2}{3}\tan^{-1}\Big(\frac{3\text{a}\text{b}^2-\text{a}^3}{\text{b}^3-3\text{a}^2\text{b}}\Big)+\frac{2}{3}\tan^{-1}\Big(\frac{3\text{x}\text{y}^2-\text{x}^3}{\text{y}^3-3\text{x}^2\text{y}}\Big)=\tan^{-1}\frac{2\alpha\beta}{\alpha^2-\beta^2}$
where $\alpha=-\text{ax}+\text{by},\beta=\text{bx}+\text{ay}$
✓

Answer

Let $\text{a}=\text{b}\tan\text{m}$ and $\text{x}=\text{y}\tan\text{n}$
Then
$\frac{2}{3}\tan^{-1}\Big(\frac{3\text{a}\text{b}^2-\text{a}^3}{\text{b}^3-3\text{a}^2\text{b}}\Big)+\frac{2}{3}\tan^{-1}\Big(\frac{3\text{x}\text{y}^2-\text{x}^3}{\text{y}^3-3\text{x}^2\text{y}}\Big)$
$=\frac{2}{3}\tan^{-1}\Big(\frac{3\text{b}^3\tan\text{m}-\text{b}^3\tan^3\text{m}}{\text{b}^3-3\text{b}^3\tan^2\text{m}}\Big)+\frac{2}{3}\tan^{-1}\Big(\frac{3\text{y}^3\tan\text{n}-\text{y}^3\tan^3\text{n}}{\text{y}^3-3\text{y}^3\tan^2\text{n}}\Big)$
$=\frac{2}{3}\tan^{-1}\Big(\frac{3\tan\text{m}-\tan^3\text{m}}{1-3\tan^2\text{m}}\Big)+\frac{2}{3}\tan^{-1}\Big(\frac{3\tan\text{n}-\tan^3\text{n}}{1-\tan^2\text{n}}\Big)$
$=\frac{2}{3}\tan^{-1}(\tan3\text{m})+\frac{2}{3}\tan^{-1}(\tan3\text{n})$ $[\because\ \text{a}=\text{b}\tan\text{m},\text{x}=\text{y}=\tan\text{n}]$
$=2\tan^{-1}\Bigg(\frac{\frac{\text{a}}{\text{b}}+\frac{\text{x}}{\text{y}}}{1-\frac{\text{a}}{\text{b}}\frac{\text{x}}{\text{y}}}\Bigg)$
$=2\tan^{-1}\Big(\frac{\text{ay}+\text{bx}}{\text{by}-\text{ax}}\Big)$
$=\tan^{-1}\begin{Bmatrix}\frac{2\frac{\text{ay}+\text{bx}}{\text{by}-\text{ax}}}{1-\Big(\frac{\text{ay}+\text{bx}}{\text{by}-\text{ax}}\Big)^2}\end{Bmatrix}$
$=\tan^{-1}\Bigg\{\frac{2(\text{ay}+\text{bx})(\text{by}-\text{ax})}{(\text{by}-\text{ax})^2-(\text{ay}+\text{bx})^2}\Bigg\}$
$=\tan^{-1}\Big\{\frac{2\alpha\beta}{\alpha^2-\beta^2}\Big\}$ $[\because\ \beta=\text{bx}+\text{ay},\alpha=-\text{ax}+\text{by}]$

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 "Solve the Following Question.(5 Marks)" from Inverse Trigonometric Functions→Inverse Trigonometric Functions — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Prove that: $\begin{vmatrix}(a+1)(a+2)&(a+2)&1\\(a+2)(a+3)&(a+3)&1\\(a+3)(a+4)&(a+4) &1\end{vmatrix}=-2$
→
$\triangle O A B$ is formed by lines $x^2-4 x y+y^2=0$ and the line $2 x+3 y-1=0$. Find the equation of the median of the triangle drawn from $\mathrm{O}$.
→
Determine if $\text{f(x)}=\begin{cases}\text{x}^2\sin\frac{1}{\text{x}},&\text{ x}\neq0\\0,&\text{x}=0\end{cases}$ is a continuous function?
→
Evaluate the following intregals:
$\int\frac{\text{x}^4}{(\text{x}-1)(\text{x}^2+1)}\ \text{dx}$
→
An unbiased die is tossed twice. Find the probability of getting 4, 5, or 6 on the first toss and 1, 2, 3 or 4 on the second toss.
→
Show that the points $\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$ and $3\hat{\text{i}}+3\hat{\text{j}}+3\hat{\text{k}}$ are equidistant from the plane $\vec{\text{r}}\cdot(5\hat{\text{i}}+2\hat{\text{j}}-7\hat{\text{k}})+9=0$
→
Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,
$f(x) = (x - 1)(x - 2)^2$​​​​​​​
→
Evaluate the following integrals as limit of sum:$\int\limits^5_{0}(\text{x}+1)\text{dx}$
→
Vitamins $A$ and $B$ are found in two different foods $F _1$ and $F _2$. One unit of food $F _1$ contains $2$ units of vitamin A and $3$ units of vitamin $B$. One unit of food $F_2$ contains $4$ units of vitamin $A$ and $2$ units of vitamin $B$. One unit of food $F_1$ and $F_2$ cost Rs $50$ and $25$ respectively. The minimum daily requirements for a person of vitamin $A$ and $B$ is $40$ and $50$ units respectively. Assuming that anything in excess of daily minimum requirement of vitamin $A$ and $B$ is not harmful, find out the optimum mixture of food $F _1$ and $F _2$ at the minimum cost which meets the daily minimum requirement of vitamin $A$ and $B$. Formulate this as a LPP.
→
Show that $\begin{vmatrix}\text{x}-3&\text{x}-4&\text{x}-\alpha\\\text{x}-2&\text{x}-3&\text{x}-\beta\\\text{x}-1&\text{x}-2&\text{x}-\gamma\end{vmatrix}=0,$ where $\alpha,\beta,\gamma$ are in A.P.
→