+ ( 3 +x )- ( 3 -x )=3 3x - Vidyadip

Question

$\tan\text{x}+\tan\big(\frac{\pi}{3}+\text{x}\big)-\tan\big(\frac{\pi}{3}-\text{x}\big)=3\tan3\text{x}$

✓

Answer

$\frac{\pi}{3}=60^\circ$
$\text{LHS}=\tan\text{x}+\tan(60^\circ+\text{x})=\tan(60^\circ-\text{x})$
$=\tan\text{x}+\big(\frac{\tan60^\circ+\tan\text{x}}{1-\tan60^\circ\tan\text{x}}\big)-\big(\frac{\tan60^\circ-\tan\text{x}}{1+\tan60^\circ\tan\text{x}}\big)$
$\Big[\tan(\text{x}+\text{y})=\frac{\tan60^\circ-\tan\text{x}}{1+\tan60^\circ\tan\text{x}}$ and $\tan(\text{(x}-\text{y})=\frac{\tan\text{x}\tan\text{y}}{1+\tan\text{x}\tan\text{y}}\Big]$
$=\tan\text{x}+\frac{\sqrt{3}+\tan\text{x}}{1-\sqrt{3}\tan\text{x}}-\frac{\sqrt{3}-\tan\text{x}}{1+\sqrt{3\tan\text{x}}}$
$=\tan\text{x}+\frac{\sqrt{3}+3\tan\text{x}+\tan\text{x}+\sqrt{3}\tan^3\text{x}+\sqrt{3}+3\tan\text{x}+\tan\text{x}-\sqrt{3}\tan^2\text{x}}{\big(1-\sqrt{3}\tan\text{x}\big)\big(1+\sqrt{3}\tan\text{x}\big)}$
$=\tan\text{x}+\frac{8\tan\text{x}}{1-3\tan^2\text{x}}$
$=\frac{\tan\text{x}-3\tan^3\text{x}+8\tan\text{x}}{1-3\tan^2\text{x}}$
$=\frac{9\tan\text{x}-3\tan^3\text{x}}{1-3\tan^2\text{x}}$
$=3\Big(\frac{3\tan\text{x}-\tan^3\text{x}}{1-3\tan^2\text{x}}\Big)\Big(\because\tan3\theta=\frac{3\tan\theta-\tan^3\theta}{1-3\tan^2\theta}\Big)$
$=3\tan3\text{x}=\text{RHS}$
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 "3 Marks Question" from Trigonometric Ratios Of Compound Angles→Trigonometric Ratios Of Compound Angles — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Express the i^–35 in the form a + ib

A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the amount spent on the postage when 8^th set of letter is mailed.

Find the sum of the following geometric series:
$\sqrt{2}+\frac{1}{\sqrt{2}}+\frac{1}{2\sqrt{2}}+\ ... \text{ to 8 terms;}$

There are 10 persons named P₁ , P₂ , P₃ , ... P_10. Out of 10 persons, 5 persons are to be arranged in a line such that in each arrangement P₁must occur whereas P₄ and P₅ do not occur. Find the number of such possible arrangements.
[Hint: Required number of arrangement = ⁷C₄ × 5!]

Find the equation of the line mid-way between the parallel lines 9x + 6y - 7 = 0 and 3x + 2y + 6 = 0.

Calculate the mean deviation about the mean of the set of first n natural numbers when n is an even number.

For the frequency distribution:

x	2	3	4	5	6	7
f	4	9	16	14	11	6

Find the standard distribution.

A beam is supported at its ends by supports which are 12 metres apart. Since the load is concentrated at its centre, there is a deflection of 3 cm at the centre and the deflected beam is in the shape of a parabola. How far from the centre is the deflection 1 cm?

A piece of equipment cost a certain factory ₹ 600,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?

If $\tan\text{x}=\frac{\text{a}}{\text{b}},$ show that $\frac{\text{x}\sin\text{x - b}\cos\text{x}}{\text{a}\sin\text{x}+\text{b}\cos\text{x}}=\frac{\text{a}^2-\text{b}^2}{\text{a}^2+\text{b}^2}.$