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