Math, asked by rithisha21, 1 year ago

prove the identities please,,,, remaining two identities ,,,,,.

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JAISAL: Theres a sina told u

Answers

Answered by MarkAsBrainliest
4

Answer :

1)

Now, cos(60° - A) cosA cos(60° + A)

= {cos(60° - A) cos(60° + A)} cosA

= (cos60° cosA + sin60° sinA) (cos60° cosA - sin60° sinA) cosA

= (cos²60° cos²A - sin²60° sin²A) cosA

= (\frac{1}{4} cos²A - \frac{3}{4} sin²A) cosA

= \frac{1}{4} (cos²A - 3 sin²A) cosA

= \frac{1}{4} (cos²A - 3 + 3 cos²A) cosA

= \frac{1}{4} (4 cos³A - 3 cosA)

= \frac{1}{4} cos3A

Hence, proved.

2)

Now, tan(60° + A) tanA tan(60° - A)

 =( \frac{tan60 + tanA}{1 - tan60 \: tanA} )(\frac{tan60 - tanA}{1 + tan60 \: tanA})tanA \\ \\ = (\frac{ {tan}^{2} 60 - {tan}^{2} A}{1 - {tan}^{2}60 \: {tan}^{2} A } )tanA \\ \\ = ( \frac{3 - {tan}^{2} A}{1 - 3 {tan}^{2}A } ) \: tanA\\ \\ = \frac{3tanA - {tan}^{3} A}{1 - 3 {tan}^{2}A } \\ \\ = tan3A

Hence, proved.

#MarkAsBrainliest


PrincessNumera: Awesome
Answered by AJAYMAHICH
0
1.


sin(a)sin(60 - a)sin(60 + a) => 
sin(a) * (sin(60)cos(a) - sin(a)cos(60)) * (sin(60)cos(a) + sin(a)cos(60)) => 
sin(a) * (sin(60)^2 * cos(a)^2 - sin(a)^2 * cos(60)^2) => 
sin(a) * ((3/4) * cos(a)^2 - (1/4) * sin(a)^2) => 
(1/4) * sin(a) * (3cos(a)^2 - sin(a)^2) => 
(1/4) * sin(a) * (3 - 3sin(a)^2 - sin(a)^2) => 
(1/4) * (3sin(a) - 4sin(a)^3) => 
(1/4) * sin(3a) 



2.

formula for cox( x+y) cos(x-y) = cos ^2 (y) - sin^2( x) 
here cos (60-A) cos (60+A) = cos^2(A)- sin^2 (60)
= cos ^2 (A) - 3/4
now cos A cos (60-A) cos (60+A = cos A ( cos^2 (A) - 3/4)


\ = 1/4 ( 4 cos ^3 (a) - 3 cos A)
{since 4 cos ^3 (a) - 3 cos A = cos 3A }
= 1/4 ( cos 3A)
hence proved 









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