Question

Verify: (sinx)/(cot²x+1)+(cosx)/(tan²x+1)=sin³)x+cos³x

Answer 1

To verify the given equation, we simplify the left-hand side and the right-hand side separately and then compare them. Both sides simplify to sin³x + cos³x, confirming that the equation is true.

Step-by-step explanation:

To verify the equation: (sinx)/(cot²x+1)+(cosx)/(tan²x+1)=sin³)x+cos³x

We will work on the left-hand side (LHS) and the right-hand side (RHS) separately and then compare them.

LHS:

Let's start by simplifying the expression on the LHS:

1. We know that cot²x+1 = csc²x
2. Similarly, tan²x+1 = sec²x
3. Substituting these values, we get:

(sinx)/(csc²x)+(cosx)/(sec²x)

Next, we can rewrite csc²x as 1/sin²x and sec²x as 1/cos²x:

(sinx)/(1/sin²x)+(cosx)/(1/cos²x)

Simplifying further, we get:

sinx*sin²x + cosx*cos²x

Using the identity sin²x + cos²x = 1, we can simplify this to:

sin³x + cos³x

RHS:

The RHS of the equation is sin³x + cos³x, which is the same as the LHS.

Therefore, the given equation is verified and true.