Question

Prove the identity of: (1 secx)(1-secx)= (tanx)(secx)?

Answer 1

To prove the trigonometric identity, we manipulate the left-hand side of the equation by factoring and using the definitions of sec(x) and tan(x). We find that (1 × sec(x))(1 - sec(x)) simplifies to the right-hand side, tan(x) × sec(x), thus proving the identity.

Step-by-step explanation:

To prove the given trigonometric identity, we start by multiplying the left-hand side terms.

(1 × sec(x))(1 - sec(x)) = 1 × (1 - sec(x)) = 1 - sec(x)

Since sec(x) is the reciprocal of cos(x), this can be rewritten as:

1 - ⅔ = 1 - ⅔ = − ⅔

Now we recall that tan(x) is equal to ⅔ by definition, which gives us:

−⅔ = tan(x) × -1

And since sec(x) is the reciprocal of cos(x), we have:

tan(x) × -1 = tan(x) × sec(x)

Thus, the original statement (1 × sec(x))(1 - sec(x)) equals to tan(x) × sec(x), proving the identity.