## Trigonometry Questions and Answers Part-11

1. $\left(\frac{1+\sin\theta-\cos\theta}{1+\sin\theta+\cos\theta}\right)^{2}$     is equal to
a) $\frac{1-\cos\theta}{1+\cos\theta}$
b) $\frac{1-\sin\theta}{1+\sin\theta}$
c) $\tan^{2}\frac{\theta}{2}$
d) Both a and c

Explanation:

2. If $\tan\theta =\frac{x^{2}+1}{x^{2}-1}$    ,then
a) $\sin\theta =\frac{x^{2}+1}{\sqrt{2\left(x^{4}+1\right)}}$
b) $\sin\theta =\frac{x^{2}-1}{\sqrt{2\left(x^{4}+1\right)}}$
c) $\cos\theta =\frac{x^{2}-1}{\sqrt{2\left(x^{4}+1\right)}}$
d) Both a and c

Explanation:

3. If $\cot\theta +\tan\theta=x$    and $\sec\theta -\cos\theta=y$    , then
a) $\sin\theta \cos\theta=\frac{1}{x}$
b) $\sin\theta \tan\theta=y$
c) $\left(x^{2}y\right)^{2/3}-\left(xy^{2}\right)^{2/3}=1$
d) All of the Above

Explanation:

4. If $\cos\alpha=\frac{3}{5}$     and $\cos\beta=\frac{5}{13}$    ,then
a) $\cos\left(\alpha-\beta\right)=\frac{63}{65}$
b) $\sin\left(\alpha+\beta\right)=\frac{56}{65}$
c) $\sin^{2}\frac{\alpha-\beta}{2}=\frac{1}{65}$
d) All of the Above

Explanation:

5. The equation $\sin^{6}x+\cos^{6}x=a^{2}$    has real solutions if
a) $a \epsilon\left(-1,1\right)$
b) $a \epsilon\left[-1,-\frac{1}{2}\right]$
c) $a \epsilon\left(-\frac{1}{2},\frac{1}{2}\right)$
d) Both b and c

Explanation:

6. If $\tan\alpha$   and $\tan\beta$   are the roots of the equation $x^{2}+px+q=0\left(p\neq 0\right)$     , then
a) $\sin^{2}\left(\alpha+\beta\right)+p \sin \left(\alpha+\beta\right)\cos\left(\alpha+\beta\right)+q\cos^{2}\left(\alpha+\beta\right)=q$
b) $\tan\left(\alpha+\beta\right)=\frac{p}{q-1}$
c) $\cos\left(\alpha+\beta\right)=1-q$
d) Both a and b

Explanation:

7. If $\sin\theta+\sin\phi=a$     and $\cos\theta+\cos\phi=b$     , then
a) $\cos\frac{\theta-\phi}{2}=\pm\frac{1}{2}\sqrt{a^{2}+b^{2}}$
b) $\cos\left(\theta-\phi\right)=\frac{a^{2}+b^{2}-2}{2}$
c) $\tan\frac{\theta-\phi}{2}=\pm\sqrt{\frac{4-a^{2}-b^{2}}{a^{2}+b^{2}}}$
d) All of the Above

Explanation:

8.If $\tan\theta=\frac{\sin\alpha-\cos\alpha}{\sin\alpha+\cos\alpha}$      , then
a) $\sin\alpha-\cos\alpha=\pm\sqrt{2}\sin\theta$
b) $\sin\alpha+\cos\alpha=\pm\sqrt{2}\cos\theta$
c) $\cos2\theta=\sin2\alpha$
d) All of the Above

Explanation:

9. Which of the following statements are possible; a, b, m and n being non-zero real numbers?
a) $4\sin^{2}\theta =5$
b) $\left(a^{2}+b^{2}\right)\cos\theta=2ab$
c) $\left(m^{2}+n^{2}\right) cosec\theta=m^{2}-n^{2}$
d) $\sin \theta =2.375$

a) $\sin 2A=-\frac{336}{625}$
b) $\cos \frac{A}{2}=\frac{\sqrt{2}}{5}$
c) $\tan \frac{A}{2}=-\frac{1}{7}$