## Trigonometry Questions and Answers Part-7

1. If $x=\sec \phi -\tan\phi$    and y = cosec $\phi +\cot\phi$    , then
a) $xy+x-y+1=0$
b) $x=\frac{y-1}{y+1}$
c) $y=\frac{1+x}{1-x}$
d) All of the Above

Explanation:

2. If tan $x=2b/\left(a-c\right)\left(a\neq c\right),y=a\cos^{2}x+2b \sin x\cos x+c\sin^{2}x$
and $z=a\sin^{2}x-2b \sin x\cos x+c\cos^{2}x$
then
a) y=z
b) y + z = a + c
c) y – z = a – c
d) Both b and c

Explanation: Adding the expression for y and z, we get

3. The values of $\theta$  lying between 0 and $\pi/2$  and satisfying the equation
$\begin{bmatrix}1+\sin ^{2}\theta & \cos^{2}\theta & 4\sin 4\theta \\\sin^{2}\theta & 1+\cos^{2}\theta & 4\sin 4\theta \\\sin^{2}\theta & \cos^{2}\theta & 1+4\sin 4\theta\end{bmatrix}=0$
are
a) $7\pi/24$
b) $5\pi/24$
c) $11\pi/24$
d) Both a and c

Explanation:

4. If $0\leq x,y\leq180^{\circ}$     and $\sin\left(x-y\right)=\cos\left(x+y\right)=1/2$      , then the values of x and y are given by
a) $x=45^{\circ}, y=15^{\circ}$
b) $x=45^{\circ}, y=135^{\circ}$
c) $x=165^{\circ}, y=135^{\circ}$
d) both a and c

Explanation:

5. If $\tan\left(x/2\right)$   = cosec x - sin x, then $\tan^{2}\left(x/2\right)$   is equal to
a) $2-\sqrt{5}$
b) $\sqrt{5}-2$
c) $\left(9-4\sqrt{5}\right)\left(2+\sqrt{5}\right)$
d) Both b and c

Explanation:

6. The value of $\cos y\cos\left(\frac{\pi}{2}-x\right)-\cos\left(\frac{\pi}{2}-y\right)\cos x+\sin y\cos\left(\frac{\pi}{2}-y\right)+\cos x\sin\left(\frac{\pi}{2}-4\right)$
is zero if
a) x = 0
b) y = 0
c) x = y
d) $x - y=\frac{3\pi}{4}$

Explanation:

7. If $\sin A,\cos A$   and $\tan A$  are in G.P, then $\cot^{6}A-\cot^{2}A$    is equal to
a) -1
b) a
c) 1
d) none of these

Explanation:

8. If $x_{i}>0$   for $1\leq i\leq n$    and $x_{1}+x_{2}+...+x_{n}=\pi$     then the greatest value of the sum $\sin x_{1}+\sin x_{2}+...+\sin x_{n}$
is equal to
a) n
b) 0
c) $n\sin\left(\frac{\pi}{n}\right)$
d) $\frac{n}{2}$

Explanation:

9. If $\sin x+\sin^{2}x+\sin^{3}x=1$     , then $\cos^{6} x-4\cos^{4}x+8\cos^{2}x$
is equal to
a) 0
b) 2
c) 4
d) 8

10. If $\sin\alpha =p,\mid p \mid\leq 1$     , then $\tan\frac{\alpha}{2},\cot\frac{\alpha}{2}$   are the roots of the equation
a) $px^{2}-2x+p=0$
b) $px^{2}-x+p=0$
c) $px^{2}+2x+p=0$
d) $px^{2}-2x-p=0$