Prove that: + 3 A - 3A = - Vidyadip

Question

Prove that:
$\frac{\sin\text{A}+\sin\text3{A}}{\cos\text{A}-\cos3\text{A}}=\cot\text{A}$

✓

Answer

We have,
$\text{LHS}=\frac{\sin\text{A}+\sin3\text{A}}{\cos\text{A}-\cos3\text{A}}$
$=\ \frac{2\sin\Big(\frac{\text{A}+3\text{A}}{2}\Big)\cos\Big(\frac{\text{A}-3\text{A}}{2}}{-2\sin\Big(\frac{\text{A}+3\text{A}}{2}\Big)\sin\Big(\frac{\text{A}-3\text{A}}{2}\Big)}$
$=\ \frac{-\sin2\text{A}\times\cos(-\text{A})}{\sin2\text{A}\sin(-\text{A})}$
$=\ \frac{-\cos(-\text{A})}{\sin(-\text{A})}$
$=\ \frac{-\cos\text{A}}{-\sin\text{A}}$ $[\because\ \cos(-\theta)=\cos\theta\text{ and }\sin(-\theta)=-\sin\theta]$
$=\ \frac{\cos\text{A}}{\sin\text{A}}$
$=\ \cot\text{A}$
$=\ \text{RHS}$
$\therefore\ \frac{\sin\text{A}+\sin3\text{A}}{\cos\text{A}-\cos3\text{A}}=\cot\text{A}.$ Hence proved.

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→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Differentiate the following from first principle

$\tan\text{2x}$

If P(15, r − 1) : P(16, r − 2) = 3 : 4, find r.

Let r and n be positive integers such that 1 < r < n. Then prove the following:
$\frac{{^\text{n}}\text{C}_{\text{r}}}{{^\text{n}}\text{C}_{\text{r}-1}}=\frac{\text{n}-\text{r}+1}{\text{r}}$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\sqrt{2}}}\frac{\sqrt{5+2\text{x}}-\big(\sqrt{2}+1\big)}{\text{x}^2-2}$

If p^th, q^th and r^th terms of an A.P. and G.P. are both a, b and c respectively, show that a^b-c b^c-a c^a-b = 1.

Prove that the lines $\text{y}=\sqrt{3}\text{x}+1,\text{y}=4$ and $\text{y}=-\sqrt{3}\text{x}+2$ form an equilateral triangle.

Find sets A, B and C such that $\text{A}\cap\text{B},\text{ A}\cap\text{C and B}\cap\text{C}$ are non-empty sets and $\text{A}\cap\text{B}\cap\text{C}=\phi.$

A point moves as so that the difference its distances from (ae, 0) and (-ae, 0) is 2 a, prove that the equation to its locus is $\frac{\text{x}^2}{\text{a}^2}-\frac{\text{y}^2}{\text{b}^2}=1 $, where $\text{b}^2=\text{a}^2(\text{e}^2-1).$

$\sin^3\text{x}+\sin^3\big(\frac{2\pi}{3}+\text{x}\big)+\sin^3\big(\frac{4\pi}{3}+\text{x}\big)=-\frac{3}{4}\sin3\text{x}$

The coefficients of 5th, 6th and 7th terms in the expansion of $(1+\text{x})^{\text{n}}$ are in A.P., find n.