Home
Class 12
MATHS
If pi<x<2pi, prove that (sqrt(1+cosx)+sq...

If `pi

Text Solution

Verified by Experts

LHS`=(sqrt(1+cosx)+sqrt(1-cosx))/(sqrt(1+cosx)-sqrt(1-cosx))`
`=(sqrt(2)|cos""(x)/(2)|+sqrt(2)|sin""(x)/(2)|)/(sqrt(2)|cos""(x)/(2)|-sqrt(2)|sin""(x)/(2)|)=(|cos""(x)/(2)|+|sin""(x)/(2)|)/(|cos""(x)/(2)|-|sin""(x)/(2)|)`
`=(-cos""(x)/(2)+sin""(x)/(2))/(-cos""(x)/(2)-sin""(x)/(2))[because piltxlt2pi,therefore(pi)/(2)lt(x)/(x)ltpi]`
Dividing numerator and enominator by `sin(x//2)`, we get
`LHS=(cot""(x)/(2)-1)/(cot""(x)/(2)+1)=cot((x)/(2)+(pi)/(4))=RHS`.
Promotional Banner

Topper's Solved these Questions

  • TRIGONOMETRIC RATIOS AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Exercise 3.1|11 Videos

  • TRIGONOMETRIC RATIOS AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Exercise 3.2|7 Videos

  • TRIGONOMETRIC FUNCTIONS

    CENGAGE|Exercise SINGLE CORRECT ANSWER TYPE|38 Videos

  • TRIGONOMETRIC RATIOS FOR COMPOUND, MULTIPLE, SUB-MULTIPLE ANGLES, AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Multiple Correct Answers Type|6 Videos

Similar Questions

Explore conceptually related problems

Evaluate lim_(xto pi)(sin(pi-x))/(pi(pi-x))

Range of tan^(-1)((2x)/(1+x^2)) is (a) [-pi/4,pi/4] (b) (-pi/2,pi/2) (c) (-pi/2,pi/4) (d) [pi/4,pi/2]

The value of sin(pi/48)cos(pi/48)cos(pi/24)cos(pi/12)cos(pi/6)cos(pi/3) is:

The function f(x)=tan^(-1)(sinx+cosx) is an increasing function in (-pi/2,pi/4) (b) (0,pi/2) (-pi/2,pi/2) (d) (pi/4,pi/2)

Prove that 32 (sqrt(3)) sin"" (pi)/(48) cos"" (pi)/(48) cos"" (pi)/(24) cos"" (pi)/(12) cos"" (pi)/(6) = 3

If f(x)=(-1)^([2x/pi]),g(x)=|sinx|-|cosx|,a n dvarphi(x)=f(x)g(x) (where [.] denotes the greatest integer function), then the respective fundamental periods of f(x),g(x),a n dvarphi(x) are a) pi,pi,pi (b) pi,2pi,pi c) pi,pi,pi/2 (d) pi,pi/2,pi

Evaluate pi_(pi/6)^(pi/3)(dx)/(1+sqrt(tanx))

Find the value of (cos .pi/7 cos . pi/17 - sin .pi/7 sin . pi/17)

The value of (cos"" (pi)/(2)+isin ""(pi)/(2))(cos ((pi)/(2^2))+isin((pi)/(2^2))) (cos ((pi)/(2^3))+isin((pi)/(2^3)))" ………."oo is