Prove that: 4A+ 3A+ 2A 4A+ 3A+ 2A = 3A - Vidyadip

Question

Prove that:
$\frac{\cos4\text{A}+\cos3\text{A}+\cos2\text{A}}{\sin4\text{A}+\sin3\text{A}+\sin2\text{A}}=\cot3\text{A}$

✓

Answer

We have,
$\text{LHS}=\frac{\cos4\text{A}+\cos3\text{A}+\cos2\text{A}}{\sin4\text{A}+\sin2\text{A}+\sin2\text{A}}$
$=\ \frac{(\cos4\text{A}+\cos2\text{A})+\cos3\text{A}}{(\sin4\text{A}+\sin2\text{A})+\sin3\text{A}}$
$=\ \frac{2\cos\Big(\frac{4\text{A}+2\text{A}}{2}\Big)\cos\Big(\frac{4\text{A}-2\text{A}}{2}\Big)+\cos3\text{A}}{2\sin\Big(\frac{4\text{A}+2\text{A}}{2}\Big)\cos\Big(\frac{4\text{A}-2\text{A}}{2}\Big)+\sin3\text{A}}$
$=\ \frac{2\cos3\text{A}\cos\text{A}+\cos3\text{A}}{2\sin3\text{A}\cos\text{A}+\sin3\text{A}}$
$=\ \frac{\cos3\text{A}(2\cos\text{A}+1)}{\sin3\text{A}(2\cos2\text{A}+1)}$
$=\ \frac{\cos3\text{A}}{\sin\text{A}}$
$=\ \cot3\text{A}$
$=\ \text{RHS}$
$\therefore\ \frac{\cos4\text{A}+\cos3\text{A}+\cos2\text{A}}{\sin4\text{A}+\sin3\text{A}+\sin2\text{A}}=\cot3\text{A}$ 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 Transformation Formulae→Transformation Formulae — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Find the number of permutations of n different things taken r at a time such that two specified things occur together?

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{{\text{x}}-\text{a}}{\sqrt{\text{x}}-\sqrt{\text{a}}}$

If $\sin\text{x}+\cos\text{x}=0$ and x lies in the fourth quadrant, find $\sin \text{x } $and $\cos\text{x}.$

Find the number of:

Diagonals.
Triangles formed in a decagon.

A rod of length 12cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with x-axis.

Let r and n be positive integers such that 1 < r < n. Then prove the following:
${^\text{n}\text{C}}_{\text{r}}+2\ {^\text{n}\text{C}}_{\text{r}-1}+{^\text{n}\text{C}}_{\text{r}-2}={^\text{n+2}\text{C}}_{\text{r}}$

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find
$(\text{A}\times\text{B})\cap(\text{A}\times\text{C})$

A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done? How many of these committees would consist of 1 man and 2 women?

Find the equation of an ellipse with its foci on y-axis, eccentricity $\frac{3}{4},$ centre at the origin and passing througth (6, 4).

An urn contains $7$ white, $5$ black and $3$ red balls. Two balls are drawn at random. Find the probability that

Both the balls are red.
One ball is red and the other is black.
One ball is white.