Answer:

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Step-by-step explanation:

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Verified answer

We can start by manipulating the given equation to get it in terms of sine and cosine:

tan a + cot a = 2

(sin a / cos a) + (cos a / sin a) = 2

(sin² a + cos² a) / (cos a sin a) = 2

(sin² a + cos² a) = 2 cos a sin a

(sin² a + cos² a) = sin 2a

Now we can use this result to find the value of sin³ a + cos³ a:

sin³ a + cos³ a = (sin a + cos a)(sin² a - sin a cos a + cos² a)

We can factor the expression sin² a + cos² a using the identity we just derived:

sin³ a + cos³ a = (sin a + cos a)(sin² a - sin a cos a + cos² a)

sin³ a + cos³ a = (sin a + cos a)(sin² a + cos² a - 2 cos a sin a)

sin³ a + cos³ a = (sin a + cos a)(1 - sin 2a)

We already know that sin² a + cos² a = 1, so we can substitute that in:

sin³ a + cos³ a = (sin a + cos a)(1 - sin 2a)

sin³ a + cos³ a = (sin a + cos a) - (sin a + cos a) sin 2a

Now we can use the original equation to substitute 2 for tan a + cot a:

sin³ a + cos³ a = (sin a + cos a) - (sin a + cos a) (sin² a + cos² a) / 2

sin³ a + cos³ a = (sin a + cos a) - (sin a + cos a) / 2

sin³ a + cos³ a = (sin a + cos a) / 2

We can use the identity (sin a + cos a)² = sin² a + 2 sin a cos a + cos² a = 1 + 2 sin a cos a to express sin a cos a in terms of (sin a + cos a)²:

sin a cos a = (1/2)(sin² a + cos² a - (sin a - cos a)²)

sin a cos a = (1/2)(1 - (sin a - cos a)²)

Now we can substitute this expression for sin a cos a in our previous result for sin³ a + cos³ a:

sin³ a + cos³ a = (sin a + cos a) / 2

sin³ a + cos³ a = [(sin a + cos a)² - (sin a - cos a)²] / 4

sin³ a + cos³ a = [(1 + 2 sin a cos a) - (1 - 2 sin a cos a)] / 4

sin³ a + cos³ a = sin a cos a

Substituting the expression we found for sin a cos a earlier:

sin³ a + cos³ a = (1/2)(sin² a + cos² a - (sin a - cos a)²)

sin³ a + cos³ a = (1/2)(1 - (sin a - cos a)²)

Therefore, the value of sin³ a + cos³ a is (1/2)(1 - (sin a - cos a)²).