Verify/Solve the identities.
1.) SIN^1/2 X COS X-SIN^5/2 X COS X
2.) Long problem, but it's fun to solve!
SEC^6 X(SEC X TAN X)-SEC^4 X(SEC X TAN X)
To verify/solve the given identities, we will simplify each expression step by step.
1.) Identity: SIN^(1/2)X COSX - SIN^(5/2)X COSX
Let's factor out common terms:
= COSX(SIN^(1/2)X - SIN^(5/2)X)
Now, let's simplify the expression within the brackets:
= COSX(SIN^(1/2)X - SIN^(2.5)X)
= COSX(SIN^(1/2)X - SIN^(5/2)X)
= COSX(SIN^(1/2)X(1 - SIN^2.5X))
Since SIN^2.5X is the same as SIN^2X * SIN^0.5X, we can rewrite the expression as:
= COSX(SIN^(1/2)X(1 - SIN^2X * SIN^0.5X))
Now, by using the Pythagorean identity SIN^2X + COS^2X = 1, we can substitute 1 - SIN^2X with COS^2X:
= COSX(SIN^(1/2)X(COS^2X * SIN^0.5X))
Next, we can simplify SIN^(1/2)X * SIN^0.5X as SINX:
= COSX(COS^2X * SINX)
Finally, we use the identity COS^2X = 1 - SIN^2X:
= COSX((1 - SIN^2X) * SINX)
= COSX(SINX - SIN^3X)
Therefore, the simplified form of the expression is SINX * COSX - SIN^3X * COSX.
2.) Identity: SEC^6X(SECX * TANX) - SEC^4X(SECX * TANX)
Let's factor out common terms:
= (SEC^6X - SEC^4X)(SECX * TANX)
Now, let's simplify the expression within the brackets:
= [(SEC^2X)^3 - (SEC^2X)^2](SECX * TANX)
Using the identity SEC^2X = 1 + TAN^2X, we can make the substitution:
= [(1 + TAN^2X)^3 - (1 + TAN^2X)^2](SECX * TANX)
Simplifying further, let's expand the binomial terms:
= [(1 + 3TAN^2X + 3TAN^4X + TAN^6X) - (1 + 2TAN^2X + TAN^4X)](SECX * TANX)
Cancel out the common terms:
= (3TAN^4X + TAN^6X - 2TAN^2X)(SECX * TANX)
Now, we can simplify the expression by combining like terms:
= TAN^2X(3TAN^2X + TAN^4X - 2)(SECX * TANX)
Therefore, the simplified form of the expression is TAN^2X(3TAN^2X + TAN^4X - 2)(SECX * TANX).