Question
Find the values:
$\tan\bigg(\sin^{-1}\frac{3}{5}+\cot^{-1}\frac{3}{2}\bigg)$
$\tan\bigg(\sin^{-1}\frac{3}{5}+\cot^{-1}\frac{3}{2}\bigg)$
✓
Answer
Putting $\sin^{-1}\frac{3}{5}=\text{x}\ \text{and}\ \cot^{-1}\frac{3}{2}=\text{x}$ so that $\sin \text{x}=\frac{3}{5}\ \text{and}\ \cot \text{y}=\frac{3}{2}$ Now, $\cos \text{x}=\sqrt{1-\sin^2\text{x}}=\sqrt{1-\frac{9}{25}}$$=\sqrt{\frac{16}{25}=\frac{4}{5}} $
And $\tan \text{x}=\frac{\sin \text{x}}{\cos \text{x}}=\frac{3}{4} \ \text{and}\tan \text{y}=\frac{1}{\cot \text{y}}=\frac{2}{3}$ $\therefore \tan\bigg(\sin^{-1}\frac{3}{5}+\cot^{-1}\frac{3}{2}\bigg)=\tan\left(\text{x}+\text{y}\right)$ $=\frac{\tan \text{x}\tan \text{y}}{1-\tan \text{x}\tan \text{y}}=\frac{\frac{3}{4}+\frac{2}{3}}{1-\frac{3}{4}\times\frac{2}{3}}=\frac{\frac{17}{12}}{\frac{1}{2}}=\frac{17}{6}$
And $\tan \text{x}=\frac{\sin \text{x}}{\cos \text{x}}=\frac{3}{4} \ \text{and}\tan \text{y}=\frac{1}{\cot \text{y}}=\frac{2}{3}$ $\therefore \tan\bigg(\sin^{-1}\frac{3}{5}+\cot^{-1}\frac{3}{2}\bigg)=\tan\left(\text{x}+\text{y}\right)$ $=\frac{\tan \text{x}\tan \text{y}}{1-\tan \text{x}\tan \text{y}}=\frac{\frac{3}{4}+\frac{2}{3}}{1-\frac{3}{4}\times\frac{2}{3}}=\frac{\frac{17}{12}}{\frac{1}{2}}=\frac{17}{6}$
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
Evaluate the integral: $\int(x+1) \sqrt{x^2+x+1} d x$
→Solve the following equation for x:
$\tan^{-1}(2+\text{x})+\tan^{-1}(2-\text{x})=\tan^{-1}\frac{2}{3},$ where $\text{x}<-\sqrt3$ or, $\text{x}>\sqrt3$
→$\tan^{-1}(2+\text{x})+\tan^{-1}(2-\text{x})=\tan^{-1}\frac{2}{3},$ where $\text{x}<-\sqrt3$ or, $\text{x}>\sqrt3$
On the set Z of integers, if the binary operation * is defined by a * b = a + b + 2, then find the identity element.
Evaluate the following integrals:$\int\frac{\text{x}\cos^{-1}\text{x}}{\sqrt{1-\text{x}^3}}\text{dx}$
→Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among 100 students, what is the probability that:
You both enter the same section?
You both enter the same section?
The corner points of the feasible region determined by the system of linear inequations are as shown below :
Answer each of the following:
i. Let $z = 13x - 15y $ be the objective function. Find the maximum and minimum values of $z$ and also the
corresponding points at which the maximum and minimum values occur.
ii. Let $z = kx + y $ be the objective function. Find $k,$ if the value of $z$ at $A$ is same as the value of $z$ at $B$.
→Answer each of the following:
i. Let $z = 13x - 15y $ be the objective function. Find the maximum and minimum values of $z$ and also the
corresponding points at which the maximum and minimum values occur.
ii. Let $z = kx + y $ be the objective function. Find $k,$ if the value of $z$ at $A$ is same as the value of $z$ at $B$.
If $\text{y}=\text{e}^{-\text{x}}\cos\text{x},$ show that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=2\text{e}^{-\text{x}}\sin\text{x}.$
→For the principal values, evaluate the following:
$\sin^{-1}\Big(-\frac{\sqrt3}{2}\Big)-2\sec^{-1}\Big(2\tan\frac{\pi}{6}\Big)$
→$\sin^{-1}\Big(-\frac{\sqrt3}{2}\Big)-2\sec^{-1}\Big(2\tan\frac{\pi}{6}\Big)$
Let $A$ and $B$ be two independent events such that $P(A) = p_1$ and $P(B) = p_2.$ Describe in words the events whose probabilities are: $p_1 + p_2 = 2p_1p_2.$
→If x and y are connected parametrically by the equation $x = a\left( {\cos t + \log \tan \frac{t}{2}} \right),$ y = a sin t, without eliminating the parameter, find $\frac{{dy}}{{dx}}$.
→