Chapter
Chapter 7
Section
Solutions 45 Videos

Prove \displaystyle \cos^4 x - \sin^4 x = 1- 2\sin^2x.

1.06mins
Q2

Prove \displaystyle \csc^2 x + \sec^2 x = \csc^2\sec^2x.

0.59mins
Q3

Prove \displaystyle \cos^2 x \cos^2 y + \sin^2x\sin^2y + \sin^2x\cos^2y + \sin^2y\cos^2x = 1.

1.31mins
Q4

Prove \displaystyle \sec^2x - \sec^2y = \tan^2x -\tan^2y.

3.23mins
Q5

Prove \displaystyle \frac{\tan x+ \tan y}{\cot x + \cot y} = (\tan x)(\tan y).

1.25mins
Q6

Prove \displaystyle (\sec x -\cos x)(\csc x - \sin x) =\frac{\tan x}{1+\tan^2x}.

2.39mins
Q7

Prove \displaystyle \cos^6x + \sin^6x = 1 - 3\sin^2x+ 3\sin^4x.

2.47mins
Q8

Prove \displaystyle \sec^6x - \tan^6x = 1 + 3\tan^2x \sec^2x.

3.42mins
Q9

Prove \displaystyle 1 + \cot x \tan y = \frac{\sin(x+ y)}{\sin x \cos y}.

1.04mins
Q10

Prove \displaystyle \cos(x+y)\cos y + \sin(x+y)\sin y = \cos x.

1.05mins
Q11

Prove \displaystyle \sin x- \tan y \cos x = \frac{\sin(x-y)}{\cos y}.

0.51mins
Q12

Prove \displaystyle \sin(x +y)\sin(x -y) = \cos^2y - \cos^2x.

2.25mins
Q13

Prove \displaystyle \tan(x+y)\tan(x-y) = \frac{\sin^2x - \sin^2y}{\cos^2x-\sin^2y} .

5.58mins
Q14

Prove \displaystyle \frac{\tan(x-y)+ \tan y}{1-\tan(x-y)\tan y} = \tan x .

1.16mins
Q15

Prove \displaystyle \sin 5x = \sin x(\cos^22x-\sin^22x) + 2\cos x \cos 2x \sin 2x.

1.42mins
Q16

Prove \displaystyle \sin(\frac{\pi}{2} - x)\cot(\frac{\pi}{2}+x) = - \sin x.

0.56mins
Q17

Prove \displaystyle \cos(-x) + \cos(\pi- x) = \cos(\pi + x)+ \cos x.

1.07mins
Q18

Prove \displaystyle \frac{\sin 2x}{1+ \cos 2x} = \tan x.

0.50mins
Q19

Prove \displaystyle \frac{1 + \cos x}{\sin x} = \cot \frac{x}{2}.

1.09mins
Q20

Prove \displaystyle 2\csc 2x = \sec x \csc x

0.47mins
Q21

Prove \displaystyle 2 \cot 2x = \cot x - \tan x

0.57mins
Q22

Prove \displaystyle \frac{\cos 2x}{1+ \sin 2x} = \tan(\frac{\pi}{4} - x)

3.20mins
Q23

Prove \displaystyle \frac{\cos x - \sin x}{\cos x + \sin x} = \sec 2x - \tan 2x

2.20mins
Q24

Prove \displaystyle \frac{1 - \cos 2x + \sin 2x}{1 + \cos 2x + \sin 2x} = \tan x

2.40mins
Q25

Prove \displaystyle \cos^6x -\sin^6x = \cos 2x(1 - \frac{1}{4}\sin^22x)

3.01mins
Q26

Prove \displaystyle 4(\cos^6x + \sin^6x) = 1 + 3\cos^2 2x

4.27mins
Q27

Prove \displaystyle \sec x - \tan x = \tan(\frac{\pi}{4}- \frac{x}{2})

4.06mins
Q28

Prove \displaystyle \frac{\sin 2x}{1+\cos 2x} \times \frac{\cos x}{1 + \cos x} = \tan \frac{x}{2}

2.46mins
Q29

Prove \displaystyle \sin^2x + \cos^4x = \cos^2 x + \sin^4 x

1.51mins
Q30

Prove \displaystyle \tan x - \cot x = (\tan x -1)(\cot x + 1)

0.34mins
Q31

Prove \displaystyle \cos x = \sin x \tan^2x\cot^3x

0.55mins
Q32

Prove \displaystyle (\sin x + \cos x)(\tan x + \cot x) = \sec x + \csc x

3.04mins
Q33

Prove \displaystyle \sin^4x + \cos^4x = \sin^2x(\csc^2x - 2\cos^2 x)

1.36mins
Q34

Prove \displaystyle \sin^3x + \cos^3x = (1- \sin x \cos x)(\sin x + \cos x)

0.52mins
Q35

Prove \displaystyle \cos(\frac{\pi}{12} -x)\sec\frac{\pi}{12} - \sin(\frac{\pi}{12} -x)\csc\frac{\pi}{12} = 4\sin x

3.48mins
Q36

Prove \displaystyle \tan(x-y)+ \tan(y - z) = \frac{\sec^2y(\tan x - \tan z)}{(1 + \tan x \tan z)(1+\tan y\tan z)}

5.59mins
Q37

Prove \displaystyle \sin 8x = 8\sin x \cos x \cos 2x \cos 4x

1.16mins
Q38

Prove \displaystyle \sin x = 1 - 2\sin^2(\frac{\pi}{4} - \frac{x}{2})

1.48mins
Q39

Prove \displaystyle \sin( x + y) + \sin (x - y) = 2\sin x \cos y

0.29mins
Q40

Prove \displaystyle \frac{\sin(x-y)}{\sin x \sin y} + \frac{\sin(y -z)}{\sin y \sin z} + \frac{\sin(z -x)}{\sin z \sin x}= 0

3.37mins
Q41

Prove \displaystyle \tan x+ \tan(\pi -x) + \cot(\frac{\pi}{2} + x) = \tan(2\pi -x)

0.58mins
Q42

Prove \displaystyle \tan(x + y + z) = \frac{\tan x \tan y \tan z - \tan x \tan y \tan z }{1 - \tan x \tan y - \tan x \tan z - \tan y \tan z}

4.56mins
Q43

Prove \displaystyle \csc^2(\frac{\pi}{2} - x) = 1 + \sin^2x\csc^2(\frac{\pi}{2} - x)

1.34mins
Q44

Prove \displaystyle \tan(\frac{\pi}{4} + x) + \tan(\frac{\pi}{4} - x) = 2\sec 2x

Prove \displaystyle \frac{1 -\sin 2x}{\cos 2x} = \frac{\cos 2x}{1+ \sin 2x}