Download App

Trigonometric Ratios of Multiple and Submultiple Angles — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Ratios of Multiple and Submultiple Angles4 Marks
Question
Prove that: $\frac{\sin\text{(A-B)}}{\cos\text{A}\cos\text{B}}+\frac{\sin\text{(B-C)}}{\cos\text{B}\cos\text{C}}+\frac{\sin\text{(C-A)}}{\cos\text{C}\cos\text{A}}=0$
✓

Answer

$\text{L.H.S}=\frac{\sin\text{(A-B)}}{\cos\text{A}\cos\text{B}}+\frac{\sin\text{(B-C)}}{\cos\text{B}\cos\text{C}}+\frac{\sin\text{(C-A)}}{\cos\text{C}\cos\text{A}}=0$ $=\frac{\sin\text{A}\cos\text{B}-\cos\text{A}\sin\text{B}}{\cos\text{A}\cos\text{B}}+\frac{\sin\text{B}\cos\text{C}-\cos\text{B}\sin\text{C}}{\cos\text{B}\cos\text{C}}\\\ \ +\frac{\sin\text{C}\cos\text{A}-\cos\text{C}\sin\text{A}}{\cos\text{C}\cos\text{A}}$ $=\frac{\sin\text{A}\cos\text{B}}{\cos\text{A}\cos\text{B}}-\frac{\cos\text{A}\sin\text{A}}{\cos\text{A}\cos\text{B}}+\frac{\sin\text{B}\cos\text{C}}{\cos\text{B}\cos\text{C}}\\\ \ -\frac{\cos\text{B}\sin\text{C}}{\cos\text{B}\cos\text{C}}+\frac{\sin\text{C}\cos\text{A}}{\cos\text{C}\cos\text{A}}-\frac{\cos\text{C}\sin\text{A}}{\cos\text{C}\cos\text{A}}$ $$$=\tan\text{A}-\tan\text{B}+\tan\text{B}-\tan\text{C}+\tan\text{C}-\tan\text{A}$ $=0$ $=\text{R.H.S}$ $\therefore\ \text{L.H.S}=\text{R.H.S}$ 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 Trigonometric Ratios of Multiple and Submultiple Angles→Trigonometric Ratios of Multiple and Submultiple Angles — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Show that in an infinite G.P. with common ratio $\text{r}\big(|\text{r}|<1\big),$ each terms bears a constant ratio to the sum of all terms that follow it.
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{\frac{\pi}{4}}}\frac{1-\sin2\text{x}}{1+\cos4\text{ x}}$
→
The following are some particulars of the distribution of weights of boys and girls in a class:
Number
Boys
Girls
  100
50
Mean weight
60kg
45kg
Variance
 9
4
Which of the distributions is more variable?
→
Solve the following equations: $\cos\text{x}+\sin\text{x}=\cos2\text{x}+\sin2\text{x}$
→
Reduce each of the following expressions to the sine and cosin of a single expression: $24\cos\text{x}+7\sin\text{x}$
→
Prove that the area of the parallelogram formed by the lines $a_1x + b_1y + c_1 = 0, a_1x + b_1y+ d_1 = 0, a_2x + b_2y + c_2 = 0, a_2x + b_2y + d_2 = 0$ is $\Big|\frac{(\text{d}_1-\text{c}_1)(\text{d}_2-\text{c}_2)}{\text{a}_1\text{b}_2-\text{a}_2\text{b}_1}\Big|$ sq.units. Deduce the condition for these lines to form a rhombus.
→
In each of the following find the equation of the hyperbola satisfying the given conditions foci $(0, \pm12), $ latus-rectum=36 [NCERT]
→
Find the equations of the circles passing through two points on y-axis at distances $3$ from the origin and having radius $5$.
→
If A, B, C are three sets such that $\text{A}\subset\text{B},$ then prove that $\text{C} - \text{B}\subset\text{C} - \text{A}.$
→
Find the rational number having the following decimal expansions:$0.\overline{68}$
→