Question

True or false? secx−cosx=sin^2x⋅secx
is an identity.



False

True

Answer 1

To determine if the equation \( \sec x - \cos x = \sin^2 x \cdot \sec x \) is an identity, we can simplify both sides and see if they are equal for all values of \( x \) where they are defined.

Recall that:
- \( \sec x = \frac{1}{\cos x} \)
- \( \sin^2 x = 1 - \cos^2 x \)

Let's rewrite both sides of the equation:

The left side:
\[
\sec x - \cos x = \frac{1}{\cos x} - \cos x
\]
To combine the terms:
\[
= \frac{1 - \cos^2 x}{\cos x} = \frac{\sin^2 x}{\cos x}
\]

Now the right side:
\[
\sin^2 x \cdot \sec x = \sin^2 x \cdot \frac{1}{\cos x} = \frac{\sin^2 x}{\cos x}
\]

Now we can see that both sides are equal:
\[
\frac{\sin^2 x}{\cos x} = \frac{\sin^2 x}{\cos x}
\]

Since both sides are equal, the assertion \( \sec x - \cos x = \sin^2 x \cdot \sec x \) is indeed an identity.

So, the answer is **True**.