Question

What basic trigonometric identity would you use to verify that sin^2x +cos^2x/cos x = sec x

Answer 1

Answer:

The basic identity used is $\bold{\sin ^{2} x+\cos ^{2} x=1}$.

Solution:

In this problem some of the basic trigonometric identities are used to prove the given expression.

Let’s first take the LHS:

$\Rightarrow \frac{\sin ^{2} x+\cos ^{2} x}{\cos x}$

Step one:

The sum of squares of Sine and Cosine is 1 which is:

$\sin ^{2} x+\cos ^{2} x=1$

On substituting the above identity in the given expression, we get,

$\Rightarrow \frac{\sin ^{2} x+\cos ^{2} x}{\cos x}=\frac{1}{\cos x} \rightarrow(1)$

Step two:

The reciprocal of cosine is secant which is:

$\cos x=\frac{1}{\sec x}$

On substituting the above identity in equation (1), we get,

$\Rightarrow \frac{\sin ^{2} x+\cos ^{2} x}{\cos x}=\sec x$

Thus, RHS is obtained.

Using the identity $\sin ^{2} x+\cos ^{2} x=1$, the given expression is verified.