Properties Derived from Trigonometric Identities

Q.

$\frac{1}{2}{\cos }^{-1}\frac{[1-x]}{[1+x]}=$

1. $co{t}^{-1}\sqrt{x}$

2. ${\tan }^{-1}\sqrt{x}$

3. ${\tan }^{-1}x$

4. $co{t}^{-1}x$

Q.

If $2\sin \theta +\tan \theta =0$, then the general values of $\theta$ are

1. $2n\pi \pm \frac{\pi }{3}$

2. $n\pi ,2n\pi \pm \frac{2\pi }{3}$

3. $n\pi ,2n\pi \pm \frac{\pi }{3}$

4. $n\pi ,n\pi +\frac{2\pi }{3}$

Q.

The maximum value of $\sin \theta +\cos \theta$ in $[\theta ,\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.]$ is

1. $\sqrt{2}$

2. $2$

3. $0$

4. $-\sqrt{2}$

Q. If 3 sin x + 5 cos x = 5, then write the value of 5 sin x − 3 cos x.

Q.

Prove the following:
$\frac{\text{cos 4x+ cos 3x + cos 2x}}{\text{sin 4x + sin 3x + sin 2x}}=\text{cot}3x$

Q.

If $x\sin \theta =y\cos \theta =\frac{2z\tan \theta }{1–{\tan }^2\theta }$, then $4z^2(x^2+y^2)$ is equal to:

1. ${\left(x^2+y^2\right)}^3$

2. ${\left(x^2-y^2\right)}^3$

3. ${\left(x^2-y^2\right)}^2$

4. ${\left(x^2+y^2\right)}^2$

Q. $ta{n}^{-1}\frac{1}{3}+ta{n}^{-1}\frac{2}{9}+ta{n}^{-1}\frac{4}{33}+\cdots$ to $\infty =$

1. $\frac{\pi }{4}$
2. $\frac{\pi }{2}$
3. $\pi$
4. $None$

Q. If sec x + tan x = k, cos x =

(a) $\frac{k^2+1}{2k}$

(b) $\frac{2k}{k^2+1}$

(c) $\frac{k}{k^2+1}$

(d) $\frac{k}{k^2-1}$

Q. The value of the expression ${\tan }^{-1}\left(\frac{\sqrt{2}}{2}\right)+{\sin }^{-1}\left(\frac{\sqrt{5}}{5}\right)-{\cos }^{-1}\left(\frac{\sqrt{10}}{10}\right),$ is

1. ${\cot }^{-1}\left(\frac{1+\sqrt{2}}{1-\sqrt{2}}\right)$
2. ${\cot }^{-1}\left(\frac{\sqrt{2}+1}{\sqrt{2}-1}\right)$
3. $-\pi +{\cot }^{-1}\left(\frac{1+\sqrt{2}}{1-\sqrt{2}}\right)$
4. $\pi -{\cot }^{-1}\left(\frac{1-\sqrt{2}}{1+\sqrt{2}}\right)$

Q.

If $x_1,x_2,x_3,x_4$ are roots of the equation $x^4-x^{3}sin2\beta +x^{2}cos2\beta -xcos\beta -sin\beta =0$ then $ta{n}^{-1}{x}_1+ta{n}^{-1}{x}_2+ta{n}^{-1}{x}_3+ta{n}^{-1}{x}_4=$

1. $\beta$
   

2. $\frac{\pi }{2}-\beta$
   

3. $\pi -\beta$
   

4. $-\beta$

Q. If $\tan x=t$ then $\tan 2x+\sec 2x=$

(a) $\frac{1+t}{1-t}$

(b) $\frac{1-t}{1+t}$

(c) $\frac{2t}{1-t}$

(d) $\frac{2t}{1+t}$

Q. The number of values of ​x in [0, 2π] that satisfy the equation ${\sin }^{2}x-\cos x=\frac{1}{4}$
(a) 1
(b) 2
(c) 3
(d) 4

Q. 11. In a triangle ABC, prove that (a) cos(A+B)+cos C=0 (b) tan (A+B)÷ 2= cot C÷ 2

Q. The value of x for which
${\tan }^{-1}x+{\sin }^{-1}x={\tan }^{-1}2x$ is

Q. If $\cot x\left(1+\sin x\right)=4m\mathrm{and}\cot x\left(1-\sin x\right)=4n,$ prove that ${\left(m^2+n^2\right)}^2=mn$.

Q. Solve the following equations:
(i) $\cos x+\cos 2x+\cos 3x=0$
(ii) $\cos x+\cos 3x-\cos 2x=0$
(iii) $\sin x+\sin 5x=\sin 3x$
(iv) $\cos x\cos 2x\cos 3x=\frac{1}{4}$
(v) $\cos x+\sin x=\cos 2x+\sin 2x$
(vi) $\sin x+\sin 2x+\sin 3=0$
(vii) $\sin x+\sin 2x+\sin 3x+\sin 4x=0$
(viii) $\sin 3x-\sin x=4{\cos }^{2}x-2$
(ix) $\sin 2x-\sin 4x+\sin 6x=0$

Q. Prove that:
(i) cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A
(ii) cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A
(iii) sin A + sin 2A + sin 4A + sin 5A = 4 cos $\frac{A}{2}$ cos $\frac{3A}{2}$ sin 3A
(iv) sin 3A + sin 2A − sin A = 4 sin A cos $\frac{A}{2}$ cos $\frac{3A}{2}$
(v) cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −$\frac{3}{4}$
(vi) $\sin \frac{\mathrm{x}}{2}\sin \frac{7x}{2}+\sin \frac{3x}{2}\sin \frac{11x}{2}=\sin 2x\sin 5x.$
(vii) $\cos x\cos \frac{x}{2}-\cos 3x\cos \frac{9x}{2}=\sin 7x\sin 8x$

Q. Show that: $2\left({\sin }^{6}x+{\cos }^{6}x\right)-3\left({\sin }^{4}x+{\cos }^{4}x\right)+1=0$

Q. If $\tan A=\frac{1}{7}$ and $\tan B=\frac{1}{3}$, show that cos 2A = sin 4B

Q. The smallest value of x satisfying the equation $\sqrt{3}\left(\cot x+\tan x\right)=4$ is
(a) $2\pi /3$
(b) $\pi /3$
(c) $\pi /6$
(d) $\pi /12$

Q. Prove the: $\left|\sqrt{{\displaystyle \frac{1-\sin x}{1+\sin x}}}\right|+\left|\sqrt{{\displaystyle \frac{1+\sin x}{1-\sin x}}}\right|=-\frac{2}{\cos x},\mathrm{where}\frac{\pi }{2}<x<\pi$

Q. If $ta{n}^{-1}(x-1)+ta{n}^{-1}(x+1)+ta{n}^{-1}x=ta{n}^{-1}3x$, then in the interval $[-1,1]$, then x has :

1. one value
2. 2 values
3. 3 values
4. No value

Q. Solve the following equations:
3sin²x – 5 sin x cos x + 8 cos² x = 2

Q. If $T_n={\sin }^{n}x+{\cos }^{n}x$, prove that

(i) $\frac{T_3-T_5}{T_1}=\frac{T_5-T_7}{T_3}$

(ii) $2T_6-3T_4+1=0$

(iii) $6T_{10}-15T_8+10T_6-1=0$

Q. $\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{\sqrt{2-\sin x}-1}{{\left({\displaystyle \frac{\mathrm{\pi }}{2}}-x\right)}^2}$

Q. $3ta{n}^{-1}\left(\frac{1}{2}\right)+2ta{n}^{-1}\left(\frac{1}{5}\right)+si{n}^{-1}\left(\frac{142}{65\sqrt{5}}\right)=$

1. $0$
2. $\pi$
3. $\frac{3\pi }{2}$
4. $2\pi$

Q. If in the triangle ABC, $\angle C=\frac{\pi }{2}$ and $si{n}^{-1}x=si{n}^{-1}\left(\frac{ax}{c}\right)+si{n}^{-1}\left(\frac{bx}{c}\right)$, where a, b, c are the sides of triangle, then total number of different values of x’ are:

1. 2
2. 3
3. 4
4. None

Q.

If $si{n}^{-1}\frac{1}{3}+si{n}^{-1}\frac{2}{3}=si{n}^{-1}x,$ then x is equal to
[Roorkee 1995]

1. \N
2. $\frac{\sqrt{5}-4\sqrt{2}}{9}$

3. $\frac{\sqrt{5}+4\sqrt{2}}{9}$

4. $\frac{x}{2}$

Q. If A + B + C = π, then sec A (cos B cos C − sin B sin C) is equal to
(a) 0
(b) −1
(c) 1
(d) None of these

Q. If $\left(2^n+1\right)x=\pi ,$ then $2^n\cos x\cos 2x\cos 2^{2}x...\cos 2^{n-1}x=1$
(a) $-1$
(b) 1
(c) 1/2
(d) None of these