Question

cos^3A+sin^3A/(sin A +cos A) + cos^3A-sin^3A/(cos A-sinA) ​

Answer 1

Given:

$\frac{ {\cos}^{3} a+ {\sin}^{3} a}{\cos \: a + \sin \: a} + \frac{ {\cos}^{3} a - {\sin}^{3} a }{\cos \: a - \sin \: a}$

To Find:

Value of above expression

Answer:

2

Explanation:

$\frac{ {\cos}^{3}a \: + \: {\sin}^{3} a}{\cos \: a + \sin \: a} + \frac{ {\cos}^{3}a - {\sin}^{3}a }{\cos \: a - \sin \: a}$

We know that:

${a}^{3} + {b}^{3} = (a + b)( {a}^{2} + {b}^{2} - ab)$

${a}^{3} - {b}^{3} =(a - b )( {a}^{2} + {b}^{2} + ab)$

$\frac{(\cos \: a +\ sin \: a)( {\cos}^{2}a \: + {\sin}^{2}a \: - \cos \: a \: \: \sin \: a) }{(\cos \: a + \sin \: a)} + \frac{(\cos \: a - \sin \: a)( {\cos}^{2}a + {\sin}^{2} a + \cos \: a \: \: sin \: a }{\cos \:a }$

We know that,

sin² a + cos² a = 1

a divided by a is 1

Cancelling (cos a+sin a) and (cos a-sin a) and performing calculation, we get:

$(1 - \cos \: a \: \:\sin \: a) + (1 + \cos \: a \: \: \sin \: a)$

$1 - \cos \: a \: \: \sin \: a + 1 + \cos \: a \: \: \sin \: a$

Cancelling cos a sin a, we get:

$2$

Additional Information:

Identities Used:

${a}^{3} + {b}^{3} = (a + b)( {a}^{2} + {b}^{2} - ab)$

${a}^{3} - {b}^{3} = (a - b) ({a}^{2} + {b}^{2} + ab)$

${\sin}^{2} a + {\cos}^{2} a = 1$

Other Trigonometric Identities:

${\sec}^{2} a - {\tan}^{2} a = 1$

${\cosec}^{2} a - {\cot}^{2} a = 1$