Trigonometric Ratios of Common Angles

Q.

Prove that: ${\sin }^{6}x+{\cos }^{6}x=1-3{\sin }^2{\cos }^{2^{}}x$

Q.

The most general value of $\theta$ satisfying the equations $\sin \theta =\sin \alpha$ and $\cos \theta =\cos \alpha$ is

1. $2n\mathrm{\pi }+\mathrm{\alpha }$

2. $2n\mathrm{\pi }-\mathrm{\alpha }$

3. $n\mathrm{\pi }+\mathrm{\alpha }$

Q. The value of $\log \tan {17}^{\circ }+\log \tan {37}^{\circ }+\log \tan {53}^{\circ }+\log \tan {73}^{\circ }$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q.

Prove that $\tan 225°cot405°+\tan 765°cot675°=0.$

Q. The angle of elevation of the top of a vertical tower from a point A, due east of it is ${45}^{\circ }$. The angle of elevation of the top of the same tower from a point B, due south of A is ${30}^{\circ }$. If the distance between A and B is $54\sqrt{2}$ m, then the height of the tower (in metres), is

1. $108$
2. $54\sqrt{3}$
3. $36\sqrt{3}$
4. $54$

Q.

If $\angle A=90°$ in the triangle ABC, then ${\tan }^{-1}\left(\frac{c}{a+b}\right)+{\tan }^{-1}\left(\frac{b}{a+c}\right)=$

1. $0$

2. $1$

3. $\frac{\mathrm{\pi }}{4}$

4. $\frac{\mathrm{\pi }}{6}$

5. $\frac{\mathrm{\pi }}{8}$

Q.

Evaluate : $\frac{\tan 65°}{\cot 25°}$

Q.

What is $\tan 180°$?

Q. If $P=\frac{\sin {300}^{\circ }\cdot \tan {330}^{\circ }\cdot \sec {420}^{\circ }}{\tan {135}^{\circ }\cdot \sin {210}^{\circ }\cdot \sec {315}^{\circ }}$ and $Q=\frac{\sec {480}^{\circ }\cdot \text{cosec}{570}^{\circ }\cdot \tan {330}^{\circ }}{\sin {600}^{\circ }\cdot \cos {660}^{\circ }\cdot \cot {405}^{\circ }},$ then the value of $P$ and $Q$ are respectively

1. $\sqrt{2},-\frac{16}{3}$
2. $\sqrt{2},\frac{16}{3}$
3. $-2,\frac{3}{16}$
4. $\sqrt{2},\frac{3}{16}$

Q.

Prove that:
(i)$\frac{cos{11}^{\circ }+sin{11}^{\circ }}{cos{11}^{\circ }-sin{11}^{\circ }}=tan{56}^{\circ }$
(ii)$\frac{cos{9}^{\circ }+sin{9}^{\circ }}{cos{9}^{\circ }-sin{9}^{\circ }}=tan{54}^{\circ }$
(iii)$\frac{cos{8}^{\circ }-sin{8}^{\circ }}{cos{8}^{\circ }+sin{8}^{\circ }}=tan{37}^{\circ }$

Q.

The value of $\tan 203°+\tan 22°+\tan 203°\tan 22°$ is

1. $1$

2. $-1$

3. $2$

4. $0$

Q. The numerical value of $\text{cosec}\theta [\frac{1-\cos \theta }{\sin \theta }+\frac{\sin \theta }{1-\cos \theta }]-2{\cot }^2\theta$ is

1. $2$
2. $3$
3. $0$
4. $1$

Q.

How do you evaluate $cot{30}^{\circ }$?

Q. The value of the expression $\frac{(1+\cos 2A)(1-\sec 2A)}{\cot A(1+\tan A)(1-\cot A)}$, when $A={30}^{\circ }$ is

Q. $ABC$ is a triangular park with $AB=AC=100$ metres. A vertical tower is sitiuated at the mid-point of $BC$. If the angle of elevation of the top of the tower at $A$ and $B$ are ${\cot }^{-1}\left(3\sqrt{2}\right)$ and ${\text{cosec}}^{-1}\left(2\sqrt{2}\right)$ respectively, then the height of the tower (in metres) is :

1. $\frac{100}{3\sqrt{3}}$
2. $25$
3. $20$
4. $10\sqrt{5}$

Q.

$\tan {70}^{°},\tan {50}^{°}+\tan {20}^{°},\tan {20}^{°}$are in

1. AP

2. GP

3. HP

4. None of these

Q. The value of $\frac{\tan {70}^{\circ }-\tan {20}^{\circ }}{\tan {50}^{\circ }}$ is

1. $2$
2. $1$
3. $0$
4. $\infty$

Q. If $\alpha ={30}^{\circ }$ and $\beta ={60}^{\circ }$, then the value of $\frac{\sin \alpha +\sec 2\alpha +\tan (\alpha +{15}^{\circ })}{\tan \beta +\cot (\frac{\beta }{2}+{15}^{\circ })+\tan \alpha }$ is

1. $\frac{7}{26}(4\sqrt{3}-3)$
2. $3(4-\sqrt{3})$
3. $\frac{1}{2}(\sqrt{3}-1)$
4. $\sqrt{3}-2$

Q. Prove that $\frac{\cos (\pi +x)\cos (-x)}{\sin (\pi -x)\cos (\frac{\pi }{2}+x)}={\cot }^{2}x$

Q. If $\cos (\alpha -\beta )=1$ and $\cos (\alpha +\beta )=\frac{1}{2},$ where $\alpha ,\beta \in (-\pi ,\pi ),$ then the number of ordered pairs $(\alpha ,\beta )$ satisfying both the equations is

1. $0$
2. $1$
3. $2$
4. $4$

Q. In any triangle ABC, prove the following:

$b\cos B+c\cos C=a\cos (B-C)$

Q. A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank is ${45}^{\circ }.$ When he moves $20\text{m}$ away from the bank, he finds the angle of elevation to be ${30}^{\circ }.$ The height of the tree is

1. $10(\sqrt{3}+1)\text{m}$
2. $15\sqrt{3}\text{m}$
3. $20(\sqrt{3}+1)\text{m}$
4. $10(\sqrt{3}-1)\text{m}$

Q. If $\sin \theta =\frac{4}{5}$ and $\theta \in [0,{90}^{\circ }]$, then the value of $\tan \theta +\cot \theta$ is

1. $\frac{25}{12}$
2. $\frac{23}{12}$
3. $\pm \frac{25}{12}$
4. $\pm \frac{23}{12}$

Q. The value of $\log \tan {17}^{\circ }+\log \tan {37}^{\circ }+\log \tan {53}^{\circ }+\log \tan {73}^{\circ }$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q. If $\sin A+\text{cosec}A=2,$ then the value of ${\sin }^{20}A+{\text{cosec}}^{20}A$ is

1. $2$
2. $20$
3. $2^{20}$
4. $1$

Q.

$\frac{1-sinAcosA}{cosA(secA-cosecA)}.\frac{si{n}^{2}A-co{s}^{2}A}{si{n}^{3}A+co{s}^{3}A}=sinA$

Q. Prove that $\frac{\tan (\frac{\pi }{4}+x)}{\tan (\frac{\pi }{4}-x)}={\left(\frac{1+\tan x}{1-\tan x}\right)}^2$

Q. $\int \frac{dx}{cos(x-a)cos(x-b)}=$

1. $cosec(a-b)log\frac{sin(x-a)}{sin(x-b)}+c$
2. $cosec(a-b)log\frac{cos(x-a)}{cos(x-b)}+c$
3. $cosec(a-b)log\frac{sin(x-b)}{sin(x-a)}+c$
4. $cosec(a-b)log\frac{cos(x-b)}{cos(x-a)}+c$

Q. Prove $\frac{\cot A\cos A}{\cot A+\cos A}=\frac{\cot A-\cos A}{\cot A\cos A}$.

Q. In triangle ABC, right angled at B, Find the expressions of cos A , sin C, tan A and cot C.

1. AB/AC , BC/AC, BC/AB , AB/BC
2. BC/AC , AB/AC, BC/AB , BC/AB
3. AB/AC , AB/AC, BC/AB , BC/AB
4. BC/AC , BC/AC, AB/BC , BC/AB