Advertisements
Advertisements
प्रश्न
Prove that (1 + sec 2θ)(1 + sec 4θ) ... (1 + sec 2nθ) = tan 2nθ
Advertisements
उत्तर
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
संबंधित प्रश्न
Find the values of sin(480°)
Find the values of cos(300°)
Find the value of the trigonometric functions for the following:
cos θ = `- 1/2`, θ lies in the III quadrant
Find the value of the trigonometric functions for the following:
cos θ = `- 2/3`, θ lies in the IV quadrant
Prove that `(cot(180^circ + theta) sin(90^circ - theta) cos(- theta))/(sin(270^circ + theta) tan(- theta) "cosec"(360^circ + theta))` = cos2θ cotθ
If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2`, find the value of cos(x − y)
If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2`, find the value of tan(x + y)
Find sin(x – y), given that sin x = `8/17` with 0 < x < `pi/2`, and cos y = `- 24/25`, x < y < `(3pi)/2`
Prove that cos(π + θ) = − cos θ
If a cos(x + y) = b cos(x − y), show that (a + b) tan x = (a − b) cot y
Express the following as a sum or difference
2 sin 10θ cos 2θ
Express the following as a product
sin 50° + sin 40°
Prove that 1 + cos 2x + cos 4x + cos 6x = 4 cos x cos 2x cos 3x
Show that cot(A + 15°) – tan(A – 15°) = `(4cos2"A")/(1 + 2 sin2"A")`
If A + B + C = 180°, prove that cos A + cos B − cos C = `- 1 + 4cos "A"/2 cos "B"/2 sin "C"/2`
If A + B + C = 180°, prove that sin A + sin B + sin C = `4 cos "A"/2 cos "B"/2 cos "C"/2`
If A + B + C = 180°, prove that sin(B + C − A) + sin(C + A − B) + sin(A + B − C) = 4 sin A sin B sin C
If ∆ABC is a right triangle and if ∠A = `pi/2` then prove that cos B – cos C = `- 1 + 2sqrt(2) cos "B"/2 sin "C"/2`
Choose the correct alternative:
`(1 + cos pi/8) (1 + cos (3pi)/8) (1 + cos (5pi)/8) (1 + cos (7pi)/8)` =
Choose the correct alternative:
cos 1° + cos 2° + cos 3° + ... + cos 179° =
