lokrajmahatara9864
Answered

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I will give 5 points
Prove the identity \frac { \sin ^ { 3 } \theta + \cos ^ { 3 } \theta } { \sin \theta + \cos \theta } = 1 - \sin \theta \cdot \cos \theta​

I Will Give 5 Points Prove The Identity Frac Sin 3 Theta Cos 3 Theta Sin Theta Cos Theta 1 Sin Theta Cdot Cos Theta class=

Sagot :

Use the sum of cubes factoring rule

[tex]a^3 + b^3 = (a+b)(a^2 - ab + b^2)[/tex]

to transform the left hand side into the right hand side.

[tex]\frac { \sin^{3} \theta + \cos^{3} \theta } { \sin \theta + \cos \theta } = 1 - \sin \theta \cdot \cos \theta\\\\\frac { (\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cdot \cos\theta + \cos^2 \theta) } { \sin \theta + \cos \theta } = 1 - \sin \theta \cdot \cos \theta\\\\[/tex]

[tex]\sin^2 \theta - \sin \theta \cdot \cos\theta + \cos^2 \theta = 1 - \sin \theta \cdot \cos \theta\\\\(\sin^2 \theta + \cos^2\theta)- \sin \theta \cdot \cos\theta = 1 - \sin \theta \cdot \cos \theta\\\\1- \sin \theta \cdot \cos\theta = 1 - \sin \theta \cdot \cos \theta \ \ \checkmark\\\\[/tex]

Throughout the entire process, the right hand side stayed the same.

On the last step, I used the pythagorean identity.

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