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RD Chapter 9- Trigonometric Ratios of Multiple and Submultiple Angles Ex-9.1 Interview Questions Answers

Question 1 : Prove the following identities:
√[(1 – cos 2x) / (1 + cos 2x)] = tan x

Answer 1 :

Let us consider LHS:

√[(1 – cos 2x) / (1 +cos 2x)]

We know that cos 2x = 1– 2 sin2 x

= 2 cos2 x – 1

So,

√[(1 – cos 2x) / (1 +cos 2x)] = √[(1 – (1 – 2sin2 x))/ (1 + (2cos2x – 1))]

= √[(1 – 1 + 2sin2 x) / (1 + 2cos2 x – 1)]

= √[2 sin2 x / 2 cos2 x]

= sin x/cos x

= tan x

= RHS

Hence proved.

Question 2 : sin 2x / (1 – cos 2x) = cot x

Answer 2 :

Let us consider LHS:

sin 2x / (1 – cos 2x)

We know that cos 2x = 1– 2 sin2 x

Sin 2x = 2 sin x cos x

So,

sin 2x / (1 – cos 2x) =(2 sin x cos x) / (1 – (1 – 2sin2 x))

= (2 sin x cos x) / (1 –1 + 2sin2 x)]

= [2 sin x cos x / 2 sin2 x]

= cos x/sin x

= cot x

= RHS

Hence proved.

Question 3 : sin 2x / (1 + cos 2x) = tan x

Answer 3 :

Let us consider LHS:

sin 2x / (1 + cos 2x)

We know that cos 2x = 1– 2 sin2 x

= 2 cos2 x – 1

Sin 2x = 2 sin x cos x

So,

sin 2x / (1 + cos 2x) =[2 sin x cos x / (1 + (2cos2x – 1))]

= [2 sin x cos x / (1 +2cos2 x – 1)]

= [2 sin x cos x / 2 cos2 x]

= sin x/cos x

= tan x

= RHS

Hence proved.

Question 4 :

Answer 4 :


Question 5 : [1 – cos 2x + sin 2x] / [1 + cos 2x + sin 2x] = tan x

Answer 5 :

Let us consider LHS:

[1 – cos 2x+ sin 2x] / [1 + cos 2x + sin 2x]

We know that, cos 2x = 1 – 2 sin2 x

= 2 cos2 x – 1

Sin 2x = 2 sin x cos x

So,

Question 6 : [sin x + sin 2x] / [1 + cos x + cos 2x] = tan x

Answer 6 :

Let us consider LHS:

[sin x + sin2x] / [1 + cos x + cos 2x]

We know that, cos 2x = cos2 x– sin2 x

Sin 2x = 2 sin x cos x

So,

= RHS

Hence proved.

Question 7 : cos 2x / (1 + sin 2x) = tan (π/4 – x)

Answer 7 :

Let us consider LHS:

cos 2x / (1 + sin 2x)

We know that, cos 2x = cos2 x – sin2 x

Sin 2x = 2 sin x cos x

So,

Question 8 : cos x / (1 – sin x) = tan (π/4 + x/2)

Answer 8 :

Let us consider LHS:

cos x / (1 – sin x)

We know that, cos 2x = cos2 x – sin2 x

Cos x = cos2 x/2 – sin2 x/2

Sin 2x = 2 sin x cos x

Sin x = 2 sin x/2 cos x/2

So,

Question 9 :

Answer 9 :


Question 10 :

Answer 10 :



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RD Chapter 9- Trigonometric Ratios of Multiple and Submultiple Angles Ex-9.1 Contributors

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