Home
Class 12
MATHS
Prove that: ((cosA+cos B)/(sinA-s inB))^...

Prove that: `((cosA+cos B)/(sinA-s inB))^n+((sinA+s inB)/(cosA-cosB))^n={2cot^n((A-B)/2),ifni se v e n0,ifni sod d,`

Text Solution

Verified by Experts

LHS`=((2cos(A+B)/(2)cos(A-B)/(2))/(2cos(A+B)/(2)sin(A-B)/(2)))^(n)`
`+((2sin(A+B)/(2)cos(A-B)/(2))/(2sin(A+B)/(2)sin(B-A)/(2)))^(n)`
`=(cot(A-B)/(2))^(n)+(-cot(A-B)/(2))^(n)`
`[because sin(-theta)=-sintheta]`
`=cot^(n)(A-B)/(2)+(-1)^(n)cot(n)(A-B)/(2)`
`=cot^(n)(A-B)/(2)[1+(-1)^(n)]`
`={0, "if n is odd" 2 cot^(n)(A-B)/(2)`, if n is even.
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

Prove that: ((cosA+cos B)/(sinA-sinB))^n+((sinA+sinB)/(cosA-cosB))^n={2cot^n((A-B)/2),ifni se v e n0,ifni sod d,

Prove that (cosA-cosB)^2+(sin A-sin B)^2=4sin^2((A-B)/2)

Show that (1+sinA)/(cosA)+(cosB)/(1-sinB)=(2sinA-2sinB)/(sin(A-B)+cosA-cosB)

Prove that: (sin5A-sin3A)/(cos5A+cos3A)=tanA (sinA+sin3A)/(cosA+cos3A)=tan2A

Prove that ((cos^3A-sin^3A)/(cosA - sinA))- ((cos^3A + sin^3A)/(cosA + sinA))= 2sin A cos A

Prove that: (sinA+s in3A+sin5A+sin7A)/(cosA+cos3A+cos5A+cos7A)=tan4A

Prove that (1+sin2A)/(cos2A)=(cosA+sinA)/(cosA-sinA)=tan((pi)/(4)+A)

If A+B+C=pi , prove that : (cosA)/(sinb sinC) + (cosB)/(sinC sin) + (cosC)/(sinA sinB) =2 .

Prove that (a^2sin(B-C))/(sinb+sinC)+(b^2"sin"(C-A))/(sinC+sinA)+(c^2"sin"(A-B))/(sinA+sinB)=0

If sinA=sinBa n dcosA=cosB , then prove that sin(A-B)/2=0