Question

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Answer 1

Answer:

hey here is your solution

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Step-by-step explanation:

$so \: here \\ \frac{sin {}^{2} \: \frac{\pi}{18} + sin {}^{2} \: \frac{\pi}{9} + sin {}^{2} \: \frac{7\pi}{18} + sin {}^{2} \: \frac{4\pi}{9} }{cos {}^{2} \: \frac{\pi}{18} + cos {}^{2} \: \frac{\pi}{9} + cos {}^{2} \: \frac{7\pi}{18} + cos {}^{2} \: \frac{4\pi}{9} } \\ \\ = \frac{sin {}^{2} . \frac{180}{18} + sin {}^{2} . \frac{180}{9} + sin {}^{2}. \frac{7 \times 180}{18} + sin {}^{2} \: \frac{4 \times 180}{9} }{cos {}^{2}. \frac{180}{18} + cos {}^{2} . \frac{180}{9} + cos {}^{2} . \frac{7 \times 180}{18} + cos {}^{2} . \frac{4 \times 180}{9} }$

$= \frac{sin {}^{2} .10 + sin {}^{2} .20 + sin {}^{2}.(7 \times 10 ) + sin {}^{2} .(4 \times 20)}{cos {}^{2}.10 + cos {}^{2}.20 + cos {}^{2} .(7 \times 10) + cos {}^{2} .(4 \times 20) } \\ \\ = \frac{sin {}^{2}.10 + sin {}^{2}.20 + sin {}^{2}.70 + sin {}^{2} .80 }{cos {}^{2}.10 + cos {}^{2} .20 + cos {}^{2} .70 + cos {}^{2}.80 }$

$so \: according \: to \: complementary \: amgle \: relation \\ we \: know \: that \\ cos \: (90 - θ) = sin \: θ \\ hence \: then \\ cos \: 10 = sin(90 - 10) = sin.80 \\ \\ cos \: 20 = sin(90 - 20) = sin.70 \\ \\ cos \: 80 = sin \: (90 - 80) = sin .10 \\ \\ cos \: 70 = sin \: (90 - 70) = sin.20$

$hence \: then \: accordingly \\ = \frac{sin {}^{2} .10 + sin {}^{2} .20 + sin {}^{2} .70 + sin {}^{2}.80 }{sin {}^{2}.80 + sin {}^{2}.70 + sin {}^{2} .20 + sin {}^{2} .10} \\ \\ = \frac{sin {}^{2} .10 + sin {}^{2} .20 + sin {}^{2} .70 + sin {}^{2} .80}{ sin {}^{2} .10 + sin {}^{2} .20 + sin {}^{2} .70 + sin {}^{2} .80 } \\ \\ = 1$

Answer 2

Question:

• Find the value of the given trignometric expression

Answer:

• The value of the trignometric expression equals 1  

Explanation:

Given expression :

$\bigstar \; \; {\underline{\boxed{\bf{ \dfrac{sin^2\dfrac{\pi }{18} + sin^2\dfrac{\pi }{9} + sin^2\dfrac{7 \pi }{18} + sin^2\dfrac{\pi }{9} }{cos^2\dfrac{\pi }{18} + cos^2\dfrac{\pi }{9} + cos^2\dfrac{7 \pi }{18} +cos^2\dfrac{\pi }{9}} }}}}$      

To Find:

• The value of the expression given above  

Required Solution:

• Simplifying the expression

★ As we know that ,  

• The value of π in radian system is 180 °

${\purple{\bigstar \; \; {\underline {\bf{ We \; can \; write \; the \; expression \; as : }}}}}$

${ : \implies } \tt \dfrac{sin^2\dfrac{\pi }{18} + sin^2\dfrac{\pi }{9} + sin^2\dfrac{7 \pi }{18} + sin^2\dfrac{4\pi }{9} }{cos^2\dfrac{\pi }{18} + cos^2\dfrac{\pi }{9} + cos^2\dfrac{7 \pi }{18} +cos^2\dfrac{4\pi }{9}} \\ \\ \\ \\ { : \implies } \tt \dfrac{sin^2\dfrac{180}{18} + sin^2\dfrac{180 }{9} + sin^2\dfrac{7 \times 180 }{18} + sin^2\dfrac{4 \times 180 }{9} }{cos^2\dfrac{180 }{18} + cos^2\dfrac{180}{9} + cos^2\dfrac{7 \times 180 }{18} +cos^2\dfrac{ 4 \times 180}{9}}$  

$\\ \\ {\purple{\bigstar \; \; {\underline {\bf{Simplifying \; the \; angles : }}}}}$

${ : \implies } \tt \dfrac{\sin^2 (10)+ \sin^2(20)+ \sin^2(7 \times 10)+ \sin^2(4 \times 20) }{\cos^2(10)+ \cos^2(20)+ \cos^2(7 \times 10) +\cos^2 ( 4 \times 20)} \\ \\ \\ \\ { : \implies } \tt \dfrac{\sin^2 (10)+ \sin^2(20)+ \sin^2(70)+ \sin^2(80}{\cos^2(10)+ \cos^2(20)+ \cos^2(70 ) +\cos^2 ( 80)}$

$\\ \\ {\purple{\bigstar \; \; {\underline {\bf{Converting \; the \; cos \; functions \; to \; sin : }}}}}$  

★  According to the Quadrant relation ,

• We know that the angles which lie in the first quadrant are positive and their cos function changes to sin and the angle would be converted to such form mentioned below

${\pink{\bigstar \; \; {\boxed{\tt{\cos ( \tt 90 -\alpha ) = \sin ( \alpha )}}}}}$

⠀⠀⠀⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━

★ Now let's convert them from cos to sin

➻ ${\pink{\bf { cos ( 90 - 80 ) = sin (80) }}}$

➻ ${\blue{\bf { cos ( 90 - 20 ) = sin ( 70) }}}$  

➻ ${\red{\bf { cos ( 90 - 20 ) = sin (20) }}}$  

➻ ${\purple{\bf { cos ( 90 - 10 ) = sin (10) }}}$  

$\\ \\ {\purple{\bigstar \; \; {\underline {\bf{Substituting \; the \; values \; we \; get : }}}}}$

${ : \implies } \tt \dfrac{\sin^2 (10)+ \sin^2(20)+ \sin^2(70)+ \sin^2(80)}{\cos^2(10)+ \cos^2(20)+ \cos^2(70 ) +\cos^2 ( 80)} \\ \\ \\ \\ { : \implies } \tt \dfrac{\sin^2 (10)+ \sin^2(20)+ \sin^2(70)+ \sin^2(80)}{\sin^2(80)+ \sin^2(70)+ \sin^2(20 ) +\sin^2 ( 10)}$

${\purple{\bigstar \; \; {\underline {\bf{Grouping \; the \; values \; we \; get : }}}}}$

${ : \implies } \tt \dfrac{\sin^2 (10)+ \sin^2(20)+ \sin^2(70)+ \sin^2(80)}{\sin^2(80)+ \sin^2(70)+ \sin^2(20 ) +\sin^2 ( 10)} \\ \\ \\ \\ { : \implies } \tt \dfrac{\sin^2 (10)+ \sin^2(20)+ \sin^2(70)+ \sin^2(80)}{\sin^2(10)+ \sin^2(20)+ \sin^2(70 ) +\sin^2 ( 80)}$    

${\purple{\bigstar \; \; {\underline {\bf{Factorising \; the \; values \; we \; get : }}}}}$  

${ : \implies } \tt \dfrac{\sin^2 (10)+ \sin^2(20)+ \sin^2(70)+ \sin^2(80)}{\sin^2(10)+ \sin^2(20)+ \sin^2(70 ) +\sin^2 ( 80)}$

${ : \implies } \tt \dfrac{sin^2 {\cancel{( 10 + 20 + 70 + 80)}} }{sin^2 \cancel{( 10 + 20 + 70 + 80 )}}$

${ : \implies } \tt \cancel\dfrac{sin^2}{sin^2}$

${ : \implies } \tt 1$    

Therefore Solved :

${\therefore \; \; {\underline{\boxed{\frak{ \dfrac{sin^2\dfrac{\pi }{18} + sin^2\dfrac{\pi }{9} + sin^2\dfrac{7 \pi }{18} + sin^2\dfrac{\pi }{9} }{cos^2\dfrac{\pi }{18} + cos^2\dfrac{\pi }{9} + cos^2\dfrac{7 \pi }{18} +cos^2\dfrac{\pi }{9}} = 1 }}} \;\bigstar}}$

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