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### AIEEE 2002

The number of solution of $$\tan \,x + \sec \,x = 2\cos \,x$$ in $$\left[ {0,\,2\,\pi } \right]$$ is
A
2
B
3
C
0
D
1

## Explanation

Given equation is $$\tan \,x + \sec \,x = 2\cos \,x$$

$$\Rightarrow$$ $${{\sin x} \over {\cos x}}$$$$+ {1 \over {\cos x}}$$ $$= 2\cos x$$

$$\Rightarrow$$ $${{\sin x + 1} \over {\cos x}} = 2\cos x$$

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

$$\Rightarrow$$ $${\sin x + 1}$$ $$= 2\left( {1 - {{\sin }^2}x} \right)$$

$$\Rightarrow$$ $$2{\sin ^2}x + \sin x - 1 = 0$$

$$\Rightarrow$$ $$\left( {2\sin x - 1} \right)\left( {1 + \sin x} \right)$$$$= 0$$

$$\Rightarrow$$ $${\sin x = {1 \over 2}}$$ and $${\sin x = - 1}$$

When $${\sin x = {1 \over 2}}$$ then possible $$x$$ = $$30^\circ$$, $$150^\circ$$

When $${\sin x = - 1}$$ then possible $$x$$ = $$270^\circ$$

So three solutions possible.
2

### AIEEE 2002

The period of $${\sin ^2}\theta$$ is
A
$${\pi ^2}$$
B
$$\pi$$
C
$$2\pi$$
D
$$\pi /2$$

## Explanation

The period of $${\sin ^2}\theta$$ is = $$\pi$$

Note :
(1) When $$n$$ is odd then the period of $${\sin ^n}\theta$$, $${\cos ^n}\theta$$, $${\csc ^n}\theta$$, $${\sec ^n}\theta$$ = $$2\pi$$

(2) When $$n$$ is even then the period of $${\sin ^n}\theta$$, $${\cos ^n}\theta$$, $${\csc ^n}\theta$$, $${\sec ^n}\theta$$ = $$\pi$$

(3) When $$n$$ is even/odd then the period of $${\tan ^n}\theta$$, $${\cot ^n}\theta$$ = $$\pi$$

(3) When $$n$$ is even/odd then the period of $$\left| {{{\sin }^n}\theta } \right|$$, $$\left| {{{\cos }^n}\theta } \right|$$, $$\left| {{{\csc }^n}\theta } \right|$$, $$\left| {{{\sec }^n}\theta } \right|$$, $$\left| {{{\tan }^n}\theta } \right|$$, $$\left| {{{\cot }^n}\theta } \right|$$ = $$\pi$$

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