Question

prove that :Cos^3 20+sin^3 10=3/4(cos20+sin10)

Answer 1

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

Answer 2

Answer:hope you understand

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