Question

\[ \text{Which of the following is incorrect?} \]

(a) \(\sin\theta=-\frac15\)
(b) \(\cos\theta=1\)
(c) \(\sec\theta=\frac12\)
(d) \(\tan\theta=20\)

Solution

We know that

\[ -1\le\sin\theta\le1 \]

\[ -1\le\cos\theta\le1 \]

So options (a) and (b) are possible.

Also,

\[ \tan\theta \]

can take any real value, so option (d) is possible.

But

\[ \sec\theta=\frac1{\cos\theta} \]

Since

\[ -1\le\cos\theta\le1 \]

therefore

\[ \sec\theta\le-1 \quad \text{or} \quad \sec\theta\ge1 \]

Hence

\[ \sec\theta=\frac12 \]

is not possible.

Answer

\[ \boxed{\sec\theta=\frac12} \]

Correct Option: (c)