Question

Prove that: 4(Cos³20°+Sin³10°)=3(Cos20°+Sin10°) ​

Answer 1

Step-by-step explanation:

We know,

4cos³x = 3cosx + cos3x

put here, x = 20°

then, 4cos³20° = 3cos20° + cos3 × 20°

= 3cos20° + cos60°

cos³20° = (3cos20° + cos60°)/4 -------(1)

similarly ,

4sin³x = 3sinx - sin3x

put here, x = 10°

4sin³10° = 3sin10° - sin30°

sin³10° = (3sin10° - sin30°)/4 -----------(2)

now,

LHS = cos³20° + sin³10°

put equations (1) and (2)

= 1/4(3cos20° + cos60°) + 1/4 ( 3sin10° - sin30°)

= 1/4( 3cos20° + cos60° + 3sin10° - sin30°)

we know,

cos60° = sin30° = 1/2

= 1/4 ( 3cos20° + 3sin10°)

= 3/4(cos20° + sin10°) = RHS

hence proved

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Answer 2

Answer:

$\huge{\underline{\mathrm\pink{\fcolorbox{white}{white}{Question:-}}}}$

Prove that: 4(Cos³20°+Sin³10°)=3(Cos20°+Sin10°)

Step-by-step explanation:

$\huge{\underline{\mathrm\pink{\fcolorbox{white}{white}{Solution:-}}}}$

$\small{\underline{\mathcal\blue{\fcolorbox{white}{white}{we Know:-}}}}$

4cos³x = 3cosx + cos3x

put here, x = 20°

then, 4cos³20° = 3cos20° + cos3 × 20°

= 3cos20° + cos60°

cos³20° = (3cos20° + cos60°)/4 -------(1)

$\small{\underline{\mathrm\red{\fcolorbox{white}{white}{similarly:-}}}}$

4sin³x = 3sinx - sin3x

put here, x = 10°

4sin³10° = 3sin10° - sin30°

sin³10° = (3sin10° - sin30°)/4 -----------(2)

$\small{\underline{\mathrm\red{\fcolorbox{white}{white}{Now:-}}}}$

LHS = cos³20° + sin³10°

put equations (1) and (2)

= 1/4(3cos20° + cos60°) + 1/4 ( 3sin10° - sin30°)

= 1/4( 3cos20° + cos60° + 3sin10° - sin30°)

$\small{\underline{\mathrm\pink{\fcolorbox{white}{white}{we know:-}}}}$

cos60° = sin30° = 1/2

= 1/4 ( 3cos20° + 3sin10°)

= 3/4(cos20° + sin10°) = RHS

$\huge{\boxed{\mathfrak\blue{\fcolorbox{white}{white}{Proved}}}}$✔️

$\huge{\boxed{\mathrm\red{\fcolorbox{white}{white}{THANKS}}}}$

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