Trigonometric Ratios for Sum of Two Angles

Q.

$\cos \left(\frac{2\mathrm{\pi }}{7}\right)+\cos \left(\frac{4\mathrm{\pi }}{7}\right)+\cos \left(\frac{6\mathrm{\pi }}{7}\right)=?$

1. Is equal to zero

2. Lies between$0\&3$

3. Is a negative number

4. Lies between $3\&6$

Q.

If $sinA=\frac{1}{2},cosB=\frac{\sqrt{3}}{2},$ where $\frac{\pi }{2}$ (i) tan(A+B)
(ii) tan(A-B)

Q.

$f\left(x\right)=\frac{\sqrt{\left|\tan x\right|+\tan x}}{\surd 3x}$ is defined for

1. $R$

2. $R-\left\{\frac{1}{3}\right\}$

3. $R-\left\{n\pi +\left(\frac{\pi }{2}\right):n\in I^{+}\right\}$

4. None of these

Q. If A+B= C, then write the value of tan A tanB tanC.

Q. The value of ${\sum }_{k=1}^{13}\frac{1}{sin(\frac{\pi }{4}+\frac{(k-1)\pi )}{6})sin(\frac{\pi }{4}+\frac{k\pi }{6})}$ is equal to

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

Q. If tanA+tanB=a and cotA+ cotB=b, prove that: cot(A+B)= $\frac{1}{a}-\frac{1}{b}$.

Q.

Differentiate the given functions w.r.t. x.

cos x cos 2x cos 3x

Q.

If ${\tan }^{-1}a+{\tan }^{-1}b={\sin }^{-1}1-{\tan }^{-1}c$ , then

1. $a+b+c=abc$

2. $ab+bc+ca=abc$

3. $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-\frac{1}{abc}=0$

4. $ab+bc+ca=a+b+c$

Q. Prove that:
(i) $tan8\theta -tan6\theta -tan2\theta =tan8\theta tan6\theta tan2\theta$
(ii) $tan{15}^{\circ }+tan{30}^{\circ }+tan{15}^{\circ }tan{30}^{\circ }=1$
(iii) $tan{36}^{\circ }+tan{9}^{\circ }+tan{36}^{\circ }tan{9}^{\circ }=1$
(iv) $tan13\theta -tan9\theta -tan4\theta =tan13\theta tan9\theta tan4\theta$

Q. If tan(A+B) =p and tan(A-B) = q, then write the value of tan2B.

Q. If tan(A+B)=p, tan(A-B)=q, then show that $tan2A=\frac{p+q}{1-pq}$.

Q. Find the value of ${\tan }^{-1}\left(1\right)+{\cos }^{-1}(-\frac{1}{2})+{\sin }^{-1}(-\frac{1}{2}).$

Q.

If $\alpha co{s}^{2}3\theta +\beta co{s}^4\theta =16co{s}^6\theta +9co{s}^2\theta$ is an identity then

1. α=1, β=18

2. α =1, β=24

3. α =3, β=24

4. α =4, β=2

Q.

Prove that $\cos 55°+\cos 65°+\cos 175°=0$.

Q. If $\tan {70}^{\circ }=\tan {20}^{\circ }+\lambda \tan {50}^{\circ },$ then $\lambda$ is equal to

1. $1$
2. $2$
3. $3$
4. $\sqrt{3}$

Q.

Evaluate :$\cos 48°$$-\sin 42°$

Q. If ${\sin }^{3}x\cos 3x+{\cos }^{3}x\sin 3x=\frac{3}{8}$, then the value of $16\sin 4x$ is

Q. If $\frac{cos(x-y)}{cos(x+y)}=\frac{m}{n}$, then write the value of tanx tany.

Q. Prove that:
(i) $\frac{sin(A+B)+sin(A-B)}{cos(A+B)+cos(A-B)}=tanA$
(ii) $\frac{sin(A-B)}{cosAcosB}+\frac{sin(B-C)}{cosBcosC}+\frac{sin(C-A)}{cosCcosA}=0$
(iii) $\frac{sin(A-B)}{sinAsinB}+\frac{sin(B-C)}{sinBsinC}+\frac{sin(C-A)}{sinCsinA}=0$
(iv) $si{n}^{2}B=si{n}^{2}A+si{n}^2(A-B)-2sinAcosBsin(A-B)$
(v) $co{s}^2+co{s}^{2}B-2cosAcosBcos(A+B)=si{n}^2(A+B)$
(vi) $\frac{tan(A+B)}{cot(A-B)}=\frac{ta{n}^{2}A-ta{n}^{2}B}{1-ta{n}^{2}Ata{n}^{2}B}$

Q. Evaluate the limit:
$\underset{x\to \frac{\pi }{4}}{lim}\frac{1-\tan x}{1-\sqrt{2}\sin x}$

Q.

The value of $\sin 16°+\cos 16°$ is

1. $\frac{1}{\sqrt{3}}\left(\sqrt{2}\cos 1°+\sin 1°\right)$

2. $\frac{1}{\sqrt{2}}\left(\cos 1°+\sqrt{3}\sin 1°\right)$

3. $\frac{1}{\sqrt{3}}\left(\cos 1°+\sqrt{2}\sin 1°\right)$

4. $\frac{1}{\sqrt{2}}\left(\sqrt{3}\cos 1°+\sin 1°\right)$

Q.

$\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6}$ is equal to

1. $\cot 7{\frac{1}{2}}^{\circ }$
2. $\sin 7{\frac{1}{2}}^{\circ }$
3. $\sin {15}^{\circ }$
4. $\cos {15}^{\circ }$

Q.

If $\alpha ,\beta$ are two different vlaues of $\theta$ lying between 0 and $2\pi$ which satisfy the equations 6 cos \theta + 8 sin\theta= 9, find the value of sin $(\alpha +\beta )$.

Q. If $tan{69}^{\circ }+tan{66}^{\circ }-tan{69}^{\circ }tan{66}^{\circ }=2k,$ then k=

1. -1
   
2. $\frac{1}{2}$
   
3. $-\frac{1}{2}$
   
4. none of these

Q. The value of $\tan \frac{\pi }{8}$ is equal to

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

Q. $$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{a.}\sin ({410}^{\circ }-A)\cos ({400}^{\circ }+A)+\cos ({410}^{\circ }-A)\sin ({400}^{\circ }+A)& \text{p. -1}\\ \text{b.}\frac{{\cos }^2{1}^{\circ }-{\cos }^2{2}^{\circ }}{2\sin 3^{\circ }\sin 1^{\circ }}\text{is equal to}& \text{q. 1}\\ \text{c.}\sin (-{870}^{\circ })+\text{cosec}(-{660}^{\circ })+\tan (-{855}^{\circ })+2\cot \left({840}^{\circ }\right)+\cos \left({480}^{\circ }\right)+\sec \left({900}^{\circ }\right)& \text{r.}\frac{1}{2}\end{array}$$

Which of the following is the CORRECT combination ?

1. $a\to p,b\to q,c\to r$
2. $a\to r,b\to q,c\to p$
3. $a\to q,b\to r,c\to p$
4. $a\to r,b\to p,c\to q$

Q.

$-\frac{2\pi }{5}$ is the principal value of

1. 
   

2. 
   

3. 
   

4. 
   

Q. $\text{If}{\tan }^{-1}x+{\tan }^{-1}y=\frac{4\pi }{5},\text{then}{\cot }^{-1}x+{\cot }^{-1}y$$\text{is equal to}$

1. $\pi$
2. $\frac{2\pi }{5}$
3. $\frac{3\pi }{5}$
4. $\frac{\pi }{5}$

Q. Prove that $\frac{\sin 5x-2\sin 3x+\sin x}{\cos 5x-\cos x}=\tan x$

Q.

Prove that $y=\frac{4sin\theta }{(2+cos\theta )}-\theta$ is an increasing on $\theta$ in $[0,\frac{\pi }{2}]$