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Class 11th

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6 months ago

Class 11th Trigonometric Functions

42. cos 6x = 32 cos⁶x – 48cos⁴x + 18 cos²x – 1

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Payal Gupta

Contributor-Level 10

42. L.H.S. = cos 6x

= cos 3 (2x)

= 4 cos³2x – 3 cos 2x [Q cos 3A = 4 cos³A – 3cos A]

= 4 [ (2 cos²x – 1)³] – 3 [ (2 cos²x – 1)] [Q cos 2x = 2 cos²x – 1]

= 4 [ (2 cos²x)³ + 3 [ (2 cos²x)² (–1) + 3 (2 cos²x) (–1)² + (–1)³] – 3 (2 cos²x) + 3

{Q (a + b)³= a³ + b³ + 3a²b + 3ab²}

= 4 [8 cos⁶x – 12 cos⁴x + 6cos²x – 1] – 6 cos²x + 3.

= 32 cos⁶x – 48 cos⁴x + 24 cos²x – 4 – 6cos²x + 3

= 32 cos⁶x – 48 cos⁴

...more

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New answer posted

6 months ago

Class 11th Trigonometric Functions

41. cos 4x = 1 – 8sin²x cos²x

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Payal Gupta

Contributor-Level 10

41. L.H.S. = cos 4x.

= cos 2 (2x)

= 1 – 2 Sin² (2x) [ cos 2x = 1 – 2 Sin²x]

= 1 – 2 [2 sin xcosx]² [ sin 2x = 2 sin xcos x]

= 1 – 2 [4 sin²xcos²x]

= 1 – 8 sin²xcos²x

= R.H.S.

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New answer posted

6 months ago

Class 11th Trigonometric Functions

40. $t a n 4 x = \frac{4 t a n x (1 - t a n^{2} x)}{1 - 6 t a n^{2} x + t a n^{4} x}$

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Payal Gupta

Contributor-Level 10

40. L.H.S. = tan 4x

We know that,

$t a n 2 = \frac{2 t a n}{1 - t a n^{2}} .$ , we can write

$L.H.S= t a n 2 (2 x) = \frac{2 t a n 2 x}{1 - t a n^{2} 2 x .}$

$= \frac{2 (\frac{2 t a n x}{1 - t a n^{2} x})}{1 - {(\frac{2 t a n x}{1 - t a n^{2} x})}^{2}}$

$= \frac{\frac{4 t a n x}{(1 - t a n^{2} x)}}{\frac{{(1 - t a n^{2} x)}^{2} - 4 t a n^{2} x}{{(1 - t a n^{2} x)}^{2}}}$

$= \frac{4 t a n x * (1 - t a n^{2} x)}{1 + t a n^{4} x - 2 t a n^{2} x - 4 t a n^{2} x}$

$= \frac{4 t a n x (1 - t a n^{2} x)}{1 - 6 t a n^{2} x + t a n^{4} x}$

= R.H.S.

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New answer posted

6 months ago

Class 11th Trigonometric Functions

39. cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1

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Payal Gupta

Contributor-Level 10

39. L.H.S. = cot x cot 2x – cot 2x cot 3x – cot 3x cot x.

= cot x cot 2x – cot 3x (cot 2x + cot x)

= cot x cot 2x – (cot 2x + cot x) [cot (2x + x)]

We know that,

$c o t (A +B) = \frac{c o tA c o tB - 1}{c o tA + c o tB}$ we can write

$L.H.S= c o t x c o t 2 x - (c o t 2 x + c o t x) [\frac{c o t 2 x \cdot c o t x - 1}{c o t 2 x + c o t x}]$

= cot x cot 2x – cot 2x cot x + 1

= R.H.S.

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New answer posted

6 months ago

Class 11th Trigonometric Functions

38. $\frac{c o s 4 x + c o s 3 x + c o s 2 x}{s i n 4 x + s i n 3 x + s i n 2 x} = c o t 3 x$

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Payal Gupta

Contributor-Level 10

38. L.H.S $= \frac{c o s 4 x + c o s 3 x + c o s 2 x}{s i n 4 x + s i n 3 x + s i n 2 x .}$

$= \frac{(c o s 4 x + c o s 2 x) + c o s 3 x}{(s i n 4 x + s i n 2 x) + s i n 3 x} .$

$= \frac{2 c o s (\frac{4 x + 2 x}{2}) c o s (\frac{4 x - 2 x}{2}) + c o s 3 x}{2 s i n (\frac{4 x + 2 x}{2}) c o s (\frac{4 x - 2 x}{2}) s i n 3 x} .$

$= \frac{2 c o s \frac{6 x}{2} c o s \frac{2 x}{2} + c o s 3 x}{2 s i n \frac{6 x}{2} c o s \frac{2 x}{2} + s i n 3 x}$

$= \frac{2 c o s 3 x c o s x + c o s 3 x}{2 s i n 3 x c o s x + 3 x}$

$= \frac{c o s 3 x}{s i n 3 x} * \frac{(2 c o s x + 1)}{(2 c o s x + 1)}$

$= c o t 3 x =R.H.S$

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New answer posted

6 months ago

Class 11th Trigonometric Functions

37. $\frac{s i n x - s i n 3 x}{s i n^{2} x - c o s^{2} x} = 2 s i n x$

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Payal Gupta

Contributor-Level 10

37. $= \frac{s i n x - s i n 3 x}{s i n^{2} x - c o s^{2} x}$

$= \frac{2 c o s (\frac{x + 3 x}{2}) s i n \frac{x - 3 x}{2}}{- (c o s^{2} x - s i n^{2} x)}$

$= \frac{2 c o s \frac{4 x}{2} s i n (- \frac{2 x}{2})}{- c o s 2 x} [? c o s 2 x = c o s^{2} x - s i n^{2} x]$

$= - \frac{2 c o s 2 x s i n x}{- c o s 2 x} [? s i n (- x) = - s i n x]$

= 2 sin x

= R.H.S.

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New answer posted

6 months ago

Class 11th Trigonometric Functions

36. $\frac{s i n x + s i n 3 x}{c o s x + c o s 3 x} = t a n 2 x$

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Payal Gupta

Contributor-Level 10

36. $= \frac{s i n x + s i n 3 x}{c o s x + c o s 3 x}$

$= \frac{2 (\frac{x + 3 x}{2}) c o s (\frac{x - 3 x}{2})}{2 c o s (\frac{x + 3 x}{2}) c o s (\frac{x - 3 x}{2})}$

$= \frac{s i n \frac{4 x}{2}}{c o s \frac{4 x}{2}}$

$= \frac{s i n 2 x}{c o s 2 x}$

$= t a n 2 x =$ R.H.S

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New answer posted

6 months ago

Class 11th Trigonometric Functions

35. $\frac{s i n x - s i n y}{c o s x + c o s y} = t a n \frac{x - y}{2}$

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Payal Gupta

Contributor-Level 10

35. L.H.S: = $L.H. S: = \frac{s i n x - s i n y}{c o s x + c o s y}$

$= \frac{2 c o s \frac{x + y}{2} s i n \frac{x - y}{2}}{2 c o s \frac{x + y}{2} c o s \frac{x - y}{2}}$

$= \frac{s i n (\frac{x - y}{2})}{c o s (\frac{x - y}{2})}$

$= t a n (\frac{x - y}{2}) = R.H.S.$ = R.H.S.

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New answer posted

6 months ago

Class 11th Trigonometric Functions

34. $\frac{s i n 5 x + s i n 3 x}{c o s 5 x + c o s 3 x} = t a n 4 x$

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Payal Gupta

Contributor-Level 10

34. $= \frac{s i n 5 x + s i n 3 x}{c o s 5 x + c o s 3 x .}$

$= \frac{2 s i n (\frac{5 x + 3 x}{2}) c o s (\frac{5 x - 3 x}{2})}{2 c o s (\frac{5 x + 3 x}{2}) c o s (\frac{5 x - 3 x}{2})}$

$= \frac{s i n \frac{8 x}{2} c o s \frac{2 x}{2}}{c o s \frac{8 x}{2} c o s \frac{2 x}{2}}$

$= \frac{s i n 4 x}{c o s 4 x} = t a n 4 x =$ R.H.S

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New answer posted

6 months ago

Class 11th Trigonometric Functions

33. $\frac{c o s 9 x - c o s 5 x}{s i n 17 x - s i n 3 x} = - \frac{s i n 2 x}{c o s 10 x}$

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Payal Gupta

Contributor-Level 10

33. $= \frac{c o s 9 x - c o s 5 x}{s i n 17 x - s i n 3 x}$

$= \frac{- 2 s i n (\frac{9 x + 5 x}{2}) s i n (\frac{9 x - 5 x}{2})}{2 c o s (\frac{17 x + 3 x}{2}) s i n (\frac{17 x - 3 x}{2})}$

$= \frac{- s i n \frac{14 x}{2} s i n \frac{4 x}{2}}{c o s \frac{20 x}{2} s i n \frac{14 x}{2}} = \frac{- s i n 7 x s i n 2 x}{c o s 10 x s i n 7 x} = \frac{- s i n 2 x}{c o s 10 x}$

= R.H.S

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