Advertisements
Advertisements
Question
Prove that (1 + sec 2θ)(1 + sec 4θ) ... (1 + sec 2nθ) = tan 2nθ
Advertisements
Solution
L.H.S (1 + sec 2θ) = `1 + 1/(cos2theta) + (cos 2theta + 1)/(cos 2theta)`
= `(2cos^2theta)/(cos 2theta)`
(1 + sec 4θ) = `1 + 1/(cos 4theta)`
= `(cos 4theta + 1)/(cos 4theta)`
= `(2 cos^2 (2theta))/(cos 4theta)`
(1 + sec 2nθ) = `1 + 1/(2^"n" theta)`
= `(cos 2^"n" theta + 1)/(2^"n" theta)`
= `(2 cos^2 2^("n" - 1) theta)/(cos 2^"n" theta)`
(1 + sec 2θ)(1 + sec 4θ) ... (1 + sec 2nθ)
= `(2^"n" cos^2 theta)/(cos 2theta) * (cos^2 2theta)/(cos 4 theta) ... (cos^2 2^("n" - 1) theta)/(cos 2^"n" theta)`
= `(2^"n" cos theta)/(cos 2^"n" theta) {cos theta* cos 2theta ... cos 2^("n" - 1) theta}`
= `(2^"n" costheta{sin 2^"n"theta})/(2^"n" sintheta cos 2^"n" theta)`
= tan 2nθ . cosθ
APPEARS IN
RELATED QUESTIONS
Find the values of cos(300°)
Find the value of the trigonometric functions for the following:
cos θ = `- 2/3`, θ lies in the IV quadrant
Find the value of the trigonometric functions for the following:
cos θ = `2/3`, θ lies in the I quadrant
Find cos(x − y), given that cos x = `- 4/5` with `pi < x < (3pi)/2` and sin y = `- 24/25` with `pi < y < (3pi)/2`
Prove that sin(π + θ) = − sin θ.
If a cos(x + y) = b cos(x − y), show that (a + b) tan x = (a − b) cot y
Show that tan 75° + cot 75° = 4
Prove that sin(A + B) sin(A – B) = sin2A – sin2B
Prove that cot(A + B) = `(cot "A" cot "B" - 1)/(cot "A" + cot "B")`
If tan x = `"n"/("n" + 1)` and tan y = `1/(2"n" + 1)`, find tan(x + y)
If θ + Φ = α and tan θ = k tan Φ, then prove that sin(θ – Φ) = `("k" - 1)/("k" + 1)` sin α
If θ is an acute angle, then find `sin (pi/4 - theta/2)`, when sin θ = `1/25`
If A + B = 45°, show that (1 + tan A)(1 + tan B) = 2
Prove that `tan (pi/4 + theta) - tan(pi/4 - theta)` = 2 tan 2θ
Prove that `32(sqrt(3)) sin pi/48 cos pi/48 cos pi/24 cos pi/12 cos pi/6` = 3
Show that `((cos theta -cos 3theta)(sin 8theta + sin 2theta))/((sin 5theta - sin theta) (cos 4theta - cos 6theta))` = 1
Prove that `(sin 4x + sin 2x)/(cos 4x + cos 2x)` = tan 3x
If A + B + C = `pi/2`, prove the following cos 2A + cos 2B + cos 2C = 1 + 4 sin A sin B sin C
Choose the correct alternative:
`1/(cos 80^circ) - sqrt(3)/(sin 80^circ)` =
Choose the correct alternative:
`(1 + cos pi/8) (1 + cos (3pi)/8) (1 + cos (5pi)/8) (1 + cos (7pi)/8)` =
