Prove that: \[\sin^2 \frac{2\pi}{5} - \sin^{2 -} \frac{\pi}{3} = \frac{\sqrt{5} - 1}{8}\]

\[\frac{2\pi}{5} = 72°, \frac{\pi}{3} = 60° \] \[LHS = \sin^2 72° - \sin^2 6°\] \[ = \sin^2 \left( 90° - 18° \right) - \frac{3}{4}\] \[ = \cos^2 18 °- \frac{3}{4} \left( \because \sin\left( 90° - \theta \right) = cos\theta \right)\] \[ = \left( \frac{\sqrt{10 + 2\sqrt{5}}}{4} \right)^2 - \frac{3}{4} \left( \because \cos18° = \frac{\sqrt{10 + 2\sqrt{5}}}{4} \right)\] \[ = \frac{10 + 2\sqrt{5}}{16} - \frac{3}{4}\] \[= \frac{10 + 2\sqrt{5} - 12}{16}\]\[ = \frac{2\sqrt{5} - 2}{16}\]\[ = \frac{\sqrt{5} - 1}{8}\]\[ = RHS\]\[\text{ Hence proved } .\]

Prove That: Sin 2 2 π 5 − Sin 2 − π 3 = √ 5 − 1 8 - Mathematics

प्रश्न

Prove that: $\sin^{2} \frac{2 π}{5} - \sin^{2 -} \frac{π}{3} = \frac{\sqrt{5} - 1}{8}$

संख्यात्मक

उत्तर

$\frac{2 π}{5} = 72 °, \frac{π}{3} = 60 °$

$L H S = \sin^{2} 72 ° - \sin^{2} 6 °$

$= \sin^{2} (90 ° - 18 °) - \frac{3}{4}$

$= \cos^{2} 18 ° - \frac{3}{4} (∵ \sin (90 ° - θ) = c o s θ)$

$= {(\frac{\sqrt{10 + 2 \sqrt{5}}}{4})}^{2} - \frac{3}{4} (∵ \cos 18 ° = \frac{\sqrt{10 + 2 \sqrt{5}}}{4})$

$= \frac{10 + 2 \sqrt{5}}{16} - \frac{3}{4}$

$= \frac{10 + 2 \sqrt{5} - 12}{16}$
$= \frac{2 \sqrt{5} - 2}{16}$
$= \frac{\sqrt{5} - 1}{8}$
$= R H S$
$Hence proved .$

shaalaa.com

Values of Trigonometric Functions at Multiples and Submultiples of an Angle

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

Q 1 Q 11 Q 2

अध्याय 9: Values of Trigonometric function at multiples and submultiples of an angle - Exercise 9.3 [पृष्ठ ४२]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 11

अध्याय 9 Values of Trigonometric function at multiples and submultiples of an angle
Exercise 9.3 | Q 1 | पृष्ठ ४२

संबंधित प्रश्न

Prove that: $\frac{\sin x + \sin 2 x}{1 + \cos x + \cos 2 x} = \tan x$

Prove that: $(\cos α + \cos β^{2}) + {(\sin α + \sin β)}^{2} = 4 \cos^{2} (\frac{α - β}{2})$

Prove that: $\cos^{3} 2 x + 3 \cos 2 x = 4 (\cos^{6} x - \sin^{6} x)$

Show that: $2 (\sin^{6} x + \cos^{6} x) - 3 (\sin^{4} x + \cos^{4} x) + 1 = 0$

Prove that: $\cos^{6} A - \sin^{6} A = \cos 2 A (1 - \frac{1}{4} \sin^{2} 2 A)$

If $\cos x = \frac{4}{5}$ and x is acute, find tan 2x

Prove that: $\cos \frac{2 π}{15} \cos \frac{4 π}{15} \cos \frac{8 π}{15} \cos \frac{16 π}{15} = \frac{1}{16}$

Prove that: $\cos \frac{π}{5} \cos \frac{2 π}{5} \cos \frac{4 π}{5} \cos \frac{8 π}{5} = \frac{- 1}{16}$

Prove that: $\cos \frac{π}{65} \cos \frac{2 π}{65} \cos \frac{4 π}{65} \cos \frac{8 π}{65} \cos \frac{16 π}{65} \cos \frac{32 π}{65} = \frac{1}{64}$

If $2 \tan α = 3 \tan β,$ prove that $\tan (α - β) = \frac{\sin 2 β}{5 - \cos 2 β}$ .

If $\sec (x + α) + \sec (x - α) = 2 \sec x$ , prove that $\cos x = \pm \sqrt{2} \cos \frac{α}{2}$

Prove that $\tan x + \tan (\frac{π}{3} + x) - \tan (\frac{π}{3} - x) = 3 \tan 3 x$

$\tan x + \tan (\frac{π}{3} + x) - \tan (\frac{π}{3} - x) = 3 \tan 3 x$

\cot x + \cot (\frac{π}{3} + x) + \cot (\frac{π}{3} - x) = 3 \cot 3 x

If $\cos 4 x = 1 + k \sin^{2} x \cos^{2} x$ , then write the value of k.

If $\frac{π}{2} < x < π,$ the write the value of $\sqrt{2 + \sqrt{2 + 2 \cos 2 x}}$ in the simplest form.

If $\frac{π}{2} < x < π$ , then write the value of $\frac{\sqrt{1 - \cos 2 x}}{1 + \cos 2 x}$ .

If $π < x < \frac{3 π}{2}$ , then write the value of $\sqrt{\frac{1 - \cos 2 x}{1 + \cos 2 x}}$ .

If $\sin α + \sin β = a and \cos α - \cos β = b then \tan \frac{α - β}{2} =$

The value of $\tan x \sin (\frac{π}{2} + x) \cos (\frac{π}{2} - x)$

$\sin^{2} (\frac{π}{18}) + \sin^{2} (\frac{π}{9}) + \sin^{2} (\frac{7 π}{18}) + \sin^{2} (\frac{4 π}{9}) =$

If $5 \sin α = 3 \sin (α + 2 β) \neq 0$ , then $\tan (α + β)$ is equal to

$2 cos x - c o s 3 x - \cos 5 x - 16 \cos^{3} x \sin^{2} x$

The value of $\tan x + \tan (\frac{π}{3} + x) + \tan (\frac{2 π}{3} + x)$ is

The value of $\frac{\sin 5 α - \sin 3 α}{\cos 5 α + 2 \cos 4 α + \cos 3 α} =$

If $tan x = \frac{a}{b}$ , then $b \cos 2 x + a \sin 2 x$

The value of $\cos \frac{π}{5} \cos \frac{2 π}{5} \cos \frac{4 π}{5} \cos \frac{8 π}{5}$ is ______.

Prove that sin 4A = 4sinA cos³A – 4 cosA sin³A

If tanθ + sinθ = m and tanθ – sinθ = n, then prove that m² – n² = 4sinθ tanθ
[Hint: m + n = 2tanθ, m – n = 2sinθ, then use m² – n² = (m + n)(m – n)]

If A lies in the second quadrant and 3tanA + 4 = 0, then the value of 2cotA – 5cosA + sinA is equal to ______.

The value of cos²48° – sin²12° is ______.

[Hint: Use cos²A – sin² B = cos(A + B) cos(A – B)]

If k = $\sin (\frac{π}{18}) \sin (\frac{5 π}{18}) \sin (\frac{7 π}{18})$ , then the numerical value of k is ______.

If tanA = $\frac{1 - \cos B}{\sin B}$ , then tan2A = ______.

Prove That: Sin 2 2 π 5 − Sin 2 − π 3 = √ 5 − 1 8 - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तर

APPEARS IN

संबंधित प्रश्न

Select a course

Prove That: Sin 2 2 π 5 − Sin 2 − π 3 = √ 5 − 1 8 - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तरShow Solution

APPEARS IN

संबंधित प्रश्न

Select a course

उत्तर