Math, asked by angel5197, 8 months ago

plz zzz help me out.... ​

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Answers

Answered by BrainlyEmpire
28

Answer:-

\displaystyle\boxed{\red{\sf\:\dfrac{\left(\:2\:\sin\:x\:-\:3\:\cos\:x\:\right)}{\left(\:4\:\sin\:x\:-\:9\:\cos\:x\:\right)}\:=\:3}}

Step-by-step-explanation:-

We have given that \sf\:\sec\:x\:=\:\dfrac{13}{5}

We have to prove that,

\displaystyle\sf\:\dfrac{\left(\:2\:\sin\:x\:-\:3\:\cos\:x\:\right)}{\left(\:4\:\sin\:x\:-\:9\:\cos\:x\:\right)}\:=\:3

Now, we know that,

\displaystyle\pink{\sf\:\sec\:x\:=\:\dfrac{1}{\cos\:x}}\sf\:\:\:-\:-\:-\:[\:Trigonometric\:identity\:]\\\\\\\implies\sf\:\dfrac{13}{5}\:=\:\dfrac{1}{\cos\:x}\\\\\\\implies\sf\:\cos\:x\:=\:\dfrac{1}{\dfrac{13}{5}}\\\\\\\implies\sf\:\cos\:x\:=\:\dfrac{1\:\times\:5}{13}\\\\\\\implies\boxed{\red{\sf\:\cos\:x\:=\:\dfrac{5}{13}}}

Now, we know that,

\displaystyle\pink{\sf\:\sin^2\:x\:+\:\cos^2\:x\:=\:1}\sf\:\:\:-\:-\:-\:[\:Trigonometric\:identity\:]\\\\\\\implies\sf\:\sin^2\:x\:+\:\left(\:\dfrac{5}{13}\:\right)^2\:=\:1\\\\\\\implies\sf\:\sin^2\:x\:=\:1\:-\:\left(\:\dfrac{5}{13}\:\right)^2\\\\\\\implies\sf\:\sin\:x\:=\:\sqrt{1\:-\:\left(\:\dfrac{5}{13}\:\right)^2}\:\:\:-\:-\:[\:Taking\:square\:roots\:]\\\\\\\implies\sf\:\sin\:x\:=\:\sqrt{\:\left(\:1\:\right)^2\:-\:\left(\:\dfrac{5}{13}\:\right)^2}\\\\\\\implies\sf\:\sin\:x\:=\:\sqrt{\left(\:1\:+\:\dfrac{5}{13}\:\right)\:\left(\:1\:-\:\dfrac{5}{13}\:\right)}\:\:\:-\:-\:[\:a^2\:-\:b^2\:=\:(\:a\:+\:b\:)\:(\:a\:-\:b\:)\:]

\\\\\\\displaystyle\implies\sf\:\sin\:x\:=\:\sqrt{\left(\:\dfrac{13\:+\:5}{13}\:\right)\:\left(\:\dfrac{13\:-\:5}{13}\:\right)}\\\\\\\implies\sf\:\sin\:x\:=\:\sqrt{\dfrac{18}{13}\:\times\:\dfrac{8}{13}}\\\\\\\implies\sf\:\sin\:x\:=\:\sqrt{\dfrac{18\:\times\:8}{13\:\times\:13}}\\\\\\\implies\sf\:\sin\:x\:=\:\sqrt{\dfrac{9\:\times\:2\:\times\:8}{13\:\times\:13}}\\\\\\\implies\sf\:\sin\:x\:=\:\sqrt{\dfrac{9\:\times\:16}{13\:\times\:13}}\\\\\\\implies\sf\:\sin\:x\:=\:\dfrac{3\:\times\:4}{13}\\\\\\\implies\boxed{\red{\sf\:\sin\:x\:=\:\dfrac{12}{13}}}

Now, by solving the LHS of the given equation, we get,

\displaystyle\sf\:LHS\:=\:\dfrac{\left(\:2\:\sin\:x\:-\:3\:\cos\:x\:\right)}{\left(\:4\:\sin\:x\:-\:9\:\cos\:x\:\right)}\\\\\\\implies\sf\:LHS\:=\:\dfrac{\left(\:2\:\times\:\dfrac{12}{13}\:-\:3\:\times\:\dfrac{5}{13}\:\right)}{\left(\:4\:\times\:\dfrac{12}{13}\:-\:9\:\times\:\dfrac{5}{13}\:\right)}\\\\\\\implies\sf\:LHS\:=\:\dfrac{\left(\:\dfrac{24}{13}\:-\:\dfrac{15}{13}\:\right)}{\left(\:\dfrac{48}{13}\:-\:\dfrac{45}{13}\:\right)}\\\\\\\implies\sf\:LHS\:=\:\dfrac{\dfrac{24\:-\:15}{13}}{\dfrac{48\:-\:45}{13}}\\\\\\\implies\sf\:LHS\:=\:\dfrac{\dfrac{9}{13}}{\dfrac{3}{13}}\\\\\\\implies\sf\:LHS\:=\:\dfrac{9}{\cancel{13}}\:\times\:\dfrac{\cancel{13}}{3}\\\\\\\implies\sf\:LHS\:=\:\cancel{\dfrac{9}{3}}\\\\\\\implies\boxed{\red{\sf\:LHS\:=\:3}}\\\\\\\implies\sf\:RHS\:=\:3\\\\\\\therefore\boxed{\red{\sf\:LHS\:=\:RHS}}

Hence proved!

Answered by Anonymous
20

Answer:

3

Explanation:

Given,

sec\theta = \frac{13}{5} \:

____________________

We know the Trigonometric identity:

tan² theta = sec² theta -1

= (13/5)² - 1

= 169/25 - 1

= (169 - 25)/25

= 144/25

tan theta = √(12/5)² = 12/5

_______________________

tan\theta = \frac{12}{5} \: \\  \\ </p><p>LHS</p><p>=</p><p>\frac{2sin\theta -3cos\theta}{4sin\theta - 9cos\theta} \:

Divide  \: numerator \:  and \:  \\  denominator \:  by  \: </p><p>cos\theta \: </p><p> ,we  \: get

= \frac{2tan\theta - 3}{4tan\theta - 9} \:  \\ </p><p>=  \\  \\ </p><p>\frac{2 \times \frac{12}{5}-3}{4 \times \frac{12}{5}-9} \:  \\  \\ </p><p>=</p><p>\frac{\frac{24-15}{5}}{\frac{48-45}{5}} \\  \\ </p><p>= </p><p>\frac{24-15}{48-45}</p><p>= </p><p>\frac{9}{3} \\  \\ </p><p>= </p><p>3 \\  \\ </p><p>= RHS

</p><p>=  \: </p><p>3 \\ </p><p>= RHS \:

hope this helps you

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