Download App

Trigonometric Functions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsTrigonometric Functions5 Marks
Question
If $\cos\text{x}=\frac{\cos\alpha+\cos\beta}{1+\cos\alpha\cos\beta}$ prove that $\tan\frac{\text{x}}{2}=\pm\tan\frac{\alpha}{2}\tan\frac{\beta}{2}$
✓

Answer

We have,
$\cos\theta=\frac{\cos\alpha+\cos\beta}{1+\cos\alpha.\cos\beta}$
Now,
$\cos\theta=\frac{1-\tan^2\frac{\theta}{2}}{1+\tan^2\frac{\theta}{2}}$
$=\frac{1-\tan^2\frac{\theta}{2}}{1+\tan^2\frac{\theta}{2}}=\frac{\cos\alpha+\cos\beta}{1+\cos\alpha\cos\beta}$
By componende and dividendo, we get
$\frac{\big(1-\tan^2\frac{\theta}{2}\big)+\big(1+\tan^2\frac{\theta}{2}\big)}{\big(1+\tan^2\frac{\theta}{2}\big)-\big(1-\tan^2\frac{\theta}{2}\big)}=\frac{1+\cos\alpha\cos\beta+\cos\alpha\cos\beta}{-(1+\cos\alpha\cos\beta-\cos\alpha\cos\beta)}$
$\frac{2}{2\tan^2\frac{\theta}{2}}=\frac{(1+\cos\alpha)(1+\cos\beta)}{(1-\cos\alpha)(1-\cos\alpha)}$
$\tan^2\frac{\theta}{2}=\frac{(1-\cos\alpha)(1-\cos\beta)}{(1+\cos\alpha)(1+\cos\alpha)}$
$=\frac{2\sin^2\frac{\alpha}{2}.1\sin^2\frac{\beta}{2}}{2\cos^2\frac{\alpha}{2}.1\cos^2\frac{\beta}{2}}$
$\tan\frac{\theta}{2}=\pm\tan\frac{\alpha}{2}.\tan\frac{\beta}{2}$

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

Similar questions

Differentiate the following from first principle$\tan\text{x}^2$
→
Solve the following systems of linear inequations graphically:
$\text{x}-\text{y}\leq1,\text{x}+2\text{y}\leq8,2\text{x}+\text{y}\geq2,\text{x}\geq0$ and $\text{y}\geq0$
→
Name the octant in which each of the following points lies:
  1. $(1, 2, 3).$
  2. $(4, -2, 3).$
  3. $(4, -2, -5).$
  4. $(4, 2, -5).$
  5. $(-4, 2, 5).$
  6. $(-3, -1, 6).$
  7. $(2, -4, -7).$
  8. $(-4, 2, -5).$
→
The water acidity in a pool is considerd normal when the average pH reading of three daily measurements is between 8.2 and 8.5. If the first two pH readings are 8.48 and 8.35, find the range of pH value for the third reading that will result in the acidity level being normal.
→
Differentiate the following from the first principle$\text{x}\sin\text{x}$
→
Let$ A = {1, 2, 4, 5}, B = {2, 3, 5, 6}, C = {4, 5, 6, 7}$. Verify the following identities:
$\text{A}-(\text{B}\cap\text{C})=(\text{A}-\text{B})\cup(\text{A}-\text{C})$
→
If sum of perpendicular distances of a variable point P (x, y) from the lines x + y - 5 = 0 and 3x - 2y + 7 = 0 is always 10. Show that P must move on a line.
→
Find the maximum and minimum values of each of the following trigonometrical expressions:
$\sin\text{x}-\cos\text{x}+1$
→
The sides of a square are $x = 6, x = 9, y = 3$ and $y = 6.$ Find the equation of a circle drawn on the diagonal of the square as its diameter.
→
If are two different valus of X lying between 0 and which satisfy the equation $6\cos\text{x}+8\sin\text{x}=9$find the value of $\sin(\alpha+\beta).$
→