Advertisements
Advertisements
प्रश्न
If `(cos^4alpha)/(cos^2beta) + (sin^4alpha)/(sin^2beta)` = 1, prove that `(cos^4beta)/(cos^2alpha) + (sin^4beta)/(sin^2alpha)` = 1
उत्तर
`(cos^4beta)/(cos^2alpha) + (sin^4beta)/(sin^2alpha) = (cos^2beta*cos^2beta)/(cos^2alpha) + (sin^2beta*sin^2beta)/(sin^2alpha)` ......(1)
Given `(cos^4alpha)/(cos^2beta) + (sin^4alpha)/(sin^2beta)` = 1
`(cos^4alpha*sin^2beta + sin^4alpha*cos^2beta)/(cos^2beta sin^2beta)` = 1
`cos^4alpha sin^2beta + sin^4alpha*cos^2beta = cos^2beta sin^2beta`
`cos^4alpha(1 - cos^2beta) + (1 - cos^2alpha)^2 cos^2beta = cos^2beta(1 - cos^2beta)`
`cos^4alpha - cos^4alpha cos^2beta + (1 - 2cos^2alpha + cos^4alpha) cos^2beta = cos^2beta - cos^4beta`
`cos^4alpha - cos^4alpha cos^2beta + cos^2beta - 2cos^2alpha cos^2beta + cos^4alpha cos^2beta = cos^2beta - cos^4beta`
`cos^4alpha - 2cos^2alpha cos^2beta + cos^4beta` = 0
`(cos^2alpha - cos^2beta)^2` = 0
`cos^2alpha = cos^2beta` ......(2)
`1 - sin^2alpha = 1 - sin^2beta`
`sin^2alpha = sin^2beta` ......(3
Using equation (2) and (3), equation (1) becomes
`(cos^4beta)/(cos^2alpha) + (sin^4beta)/(sin^2alpha) = (cos^2beta*cos^2beta)/(cos^2alpha) + (sin^2beta*sin^2beta)/(sin^2alpha)`
= `(cos^2beta*cos^2alpha)/(cos^2alpha) + (sin^2beta*sin^2alpha)/(sin^2alpha)`
= `cos^2beta + sin^2beta`
= 1
APPEARS IN
संबंधित प्रश्न
Identify the quadrant in which an angle given measure lies
25°
Identify the quadrant in which an angle given measure lies
825°
Identify the quadrant in which an angle given measure lies
– 55°
Identify the quadrant in which an angle given measure lies
328°
Identify the quadrant in which an angle given measure lies
– 230°
For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
1150°
For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
– 270°
For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
– 450°
If a cos θ − b sin θ = c, show that a sin θ + b cos θ = `+- sqrt("a"^2 + "b"^2 - "c"^2)`
If sin θ + cos θ = m, show that cos6θ + sin6θ = `(4 - 3("m"^2 - 1)^2)/4`, where m2 ≤ 2
If y = `(2sinalpha)/(1 + cosalpha + sinalpha)`, then prove that `(1 - cosalpha + sinalpha)/(1 + sinalpha)` = y
If x = `sum_("n" = 0)^oo cos^(2"n") theta, y = sum_("n" = 0)^oo sin^(2"n") theta` and z = `sum_("n" = 0)^oo cos^(2"n") theta, sin^(2"n") theta, 0 < theta < pi/2`, then show that xyz = x + y + z. [Hint: Use the formula 1 + x + x2 + x3 + . . . = `1/(1 - x), where |x| < 1]
If sec θ + tan θ = p, obtain the values of sec θ, tan θ and sin θ in terms of p
If cot θ(1 + sin θ) = 4m and cot θ(1 – sin θ) = 4n then prove that (m2 – n2)2 = m
Eliminate θ from the equations a sec θ – c tan θ = b and b sec θ + d tan θ = c
Choose the correct alternative:
Which of the following is not true?
