Verify the identities algebraically.
1.) TAN^5 X= TAN³X SEC²X-TAN³X
2.) COS³X SIN²X= (SIN²X-SIN^4X)COS X
LS = cosx (cos^2 x) (sin^2 x)
= cosx (1 - sin^2 x) (sin^2 x)
= cosx(sin^2 x - sin^4 x)
= RS
To verify the given identities algebraically, we need to convert both sides of the equation to the same form and simplify until they are equal. Let's start with the first identity:
1.) TAN^5X = TAN³X SEC²X - TAN³X
First, let's simplify the right-hand side (RHS) of the equation:
TAN³X SEC²X - TAN³X
Using the identity SEC²X = 1 + TAN²X, we substitute this value:
TAN³X (1 + TAN²X) - TAN³X
Expanding:
TAN³X + TAN⁵X - TAN³X
Simplifying:
TAN⁵X
Now, we have TAN^5X on the RHS, which is equivalent to the LHS of the equation. Hence, the first identity is verified algebraically.
Moving on to the second identity:
2.) COS³X SIN²X = (SIN²X - SIN^4X) COS X
Let's simplify both sides of the equation:
LHS: COS³X SIN²X
RHS: (SIN²X - SIN^4X) COS X
Expanding the right-hand side:
LHS: COS³X SIN²X
RHS: SIN²X COS X - SIN^4X COS X
Using the Pythagorean identity SIN²X = 1 - COS²X, we substitute this value in:
LHS: COS³X SIN²X
RHS: (1 - COS²X) COS X - (1 - COS²X)² COS X
Expanding further:
LHS: COS³X SIN²X
RHS: COS X - COS³X - COS X + 2 COS³X - COS X COS²X + 2 COS³X COS²X - COS X COS⁴X - 2 COS³X COS⁴X + COS²X COS⁴X
Simplifying the RHS:
LHS: COS³X SIN²X
RHS: 4 COS³X - 2 COS X - 2 COS³X COS²X - 2 COS³X COS⁴X + COS²X COS⁴X
Comparing the LHS and RHS, we have:
COS³X SIN²X = 4 COS³X - 2 COS X - 2 COS³X COS²X - 2 COS³X COS⁴X + COS²X COS⁴X
The equation holds true. Hence, the second identity is verified algebraically.
Note: In some cases, it is possible to simplify both sides of the equation further or use alternative trigonometric identities to reach the same conclusion.