Download App

Transformation Formulae — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTransformation Formulae4 Marks
Question
Prove that: $\frac{\sin\text{A}\sin2\text{A}+\sin3\text{A}\sin6\text{A}}{\sin\text{A}\cos2\text{A}+\sin3\text{A}\cos6\text{A}}=\tan5\text{A}$
✓

Answer

We have, $\text{LHS}=\frac{\sin\text{A}\sin2\text{A}+\sin3\text{A}\sin6\text{A}}{\sin\text{A}\cos2\text{A}+\sin3\text{A}\cos6\text{A}}$ $=\ \frac{2[\sin\text{A}\sin2\text{A}+\sin3\text{A}\sin6\text{A}]}{2[\sin\text{A}\cos2\text{A}+\sin3\text{A}\cos6\text{A}]}$ $=\ \frac{2\sin2\text{A}\sin\text{A}+2\sin6\text{A}\sin3\text{A}}{2\cos2\text{A}\sin\text{A}+2\cos6\text{A}\sin3\text{A}}$ $=\ \frac{\cos(2\text{A}-\text{A})-\cos(2\text{A}+\text{A})+\cos(6\text{A}-3\text{A})-\cos(6\text{A}+3\text{A})}{\sin(2\text{A}+\text{A})-\sin(2\text{A}-\text{A})+\sin(6\text{A}+3\text{A})+\sin(6\text{A}-3\text{A})}$ $=\ \frac{\cos\text{A}-\cos3\text{A}+\cos3\text{A}-\cos9\text{A}}{\sin3\text{A}-\sin\text{A}+\sin9\text{A}-\sin3\text{A}}$ $=\ \frac{\cos\text{A}-\cos9\text{A}}{\sin9\text{A}-\sin\text{A}}$ $=\ \frac{-[\cos9\text{A}-\cos\text{A}]}{\sin9\text{A}-\sin\text{A}}$ $=\ \frac{-\Big(-2\sin\Big(\frac{9\text{A}+\text{A}}{2}\Big)\times\sin\Big(\frac{9\text{A}-\text{A}}{2}\Big)\Big)}{2\sin\Big(\frac{9\text{A}-\text{A}}{2}\Big)\times\cos\Big(\frac{9\text{A}+\text{A}}{2}\Big)}$ $=\ \frac{\sin5\text{A}\sin4\text{A}}{\sin4\text{A}\cos5\text{A}}$ $=\ \tan5\text{A}$ $=\ \text{RHS}$ $\therefore\ \frac{\sin\text{A}\sin2\text{A}+\sin3\text{A}\sin6\text{A}}{\sin\text{A}\cos2\text{A}+\sin3\text{A}\cos6\text{A}}=\tan5\text{A}$ 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 "(Each question 4 marks)" from Transformation Formulae→Transformation Formulae — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Find $\sin\frac{\text{x}}{2},\cos\frac{\text{x}}{2}$ and $\tan\frac{\text{x}}{2}$ in each of the following : $\tan\text{x}=-\frac{4}{3},$ x in quadrant II
→
Differentiate the following from first principle:$(\text{x}+2)^3$
→
In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?
→
What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these are face cards.
→
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. $\frac{\text{x}^2}{16}+\frac{\text{y}^2}{9}=1$
→
In an examination, a student has to answer 4 questions out of 5 questions 1 and 2 are however compulsory. Determine the number of ways in which the student can make the choice.
→
If $\alpha$ and $\beta$ are different complex numbers with $|\beta|=1,$ then find $\Big|\frac{\beta-\alpha}{1-\bar{\alpha}\beta}\Big|.$
→
If $S_1, S_2, S_3$, be respectively the sums of n, 2n, 3n terms of a G.P., then prove that $\text{S}^2_1+\text{S}^2_2=\text{S}_1(\text{S}_2+\text{S}_3).$
→
Find the equation of the circle which circumscribes the triangle formed by the lines $y = x + 2, 3y = 4x$ and $2y = 3x.$
→
Find the conditions that the straight lines $y = m_1x + c_1, y = m_2x + c_2$ and $y = m_3x + c_3$ may meet in a point.
→