Answer
$$\sec(\pi-x)=-\sec x$$
The equation has been verified to be an identity as below.
Work Step by Step
$$\sec(\pi-x)=-\sec x$$
The left side should be rewritten first as it is more complex.
$$X=\sec(\pi-x)$$
- From reciprocal identity: $$\sec\theta=\frac{1}{\cos\theta}$$
Therefore, $$X=\frac{1}{\cos(\pi-x)}$$
- From difference identity for cosine: $\cos(A-B)=\cos A\cos B+\sin A\sin B$
Apply the identity for $\cos(\pi-x)$:
$$X=\frac{1}{\cos \pi\cos x+\sin \pi\sin x}$$
$$X=\frac{1}{-1\times\cos x+0\times\sin x}$$
$$X=\frac{1}{-\cos x}$$
$$X=-\frac{1}{\cos x}$$
Apply back the reciprocal identity mentioned in the beginning to rewrite $\frac{1}{\cos x}$ into $\sec x$.
$$X=-\sec x$$
So, it has been proved that $$\sec(\pi-x)=-\sec x$$ and the equation is an identity.
