Advertisements
Advertisements
प्रश्न
If A + B + C = 180°, prove that cos A + cos B − cos C = `- 1 + 4cos "A"/2 cos "B"/2 sin "C"/2`
Advertisements
उत्तर
(cos A + cos B) − cos C = `2 cos ("A" + "B")/2 cos ("A" - "B")/2- [1 - 2 sin^2 "C"/2]`
Hint: `[cos ("A" + "B")/2 = sin "C"/2]`
= `2sin "C"/2 cos ("A" - "B")/2 - 1 + 2sin^2 "C"/2`
= `- 1 + 2sin "C"/2[cos ("A" - "B")/2 + sin "C"/2]`
= `- 1 + 2 sin "C"/2[cos ("A" - "B")/2 + cos ("A" + "B")/2]`
= `- 1 + 2sin "C"/2[2cos (2"A")/4 + cos (2"B")/4]`
= `- 1 + 2sin "C"/2[2cos "A"/2 cos "B"/2]`
= `- 1 + 4 cos "A"/2 cos "B"/2 sin "C"/2`
= R.H.S
APPEARS IN
संबंधित प्रश्न
`(5/7, (2sqrt(6))/7)` is a point on the terminal side of an angle θ in standard position. Determine the six trigonometric function values of angle θ
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 A = `3/5` and cos B = `9/41 0 < "A" < pi/2, 0 < "B" < pi/2`, find the value of sin(A + B)
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`
Expand cos(A + B + C). Hence prove that cos A cos B cos C = sin A sin B cos C + sin B sin C cos A + sin C sin A cos B, if A + B + C = `pi/2`
If a cos(x + y) = b cos(x − y), show that (a + b) tan x = (a − b) cot y
Prove that sin(n + 1) θ sin(n – 1) θ + cos(n + 1) θ cos(n – 1)θ = cos 2θ, n ∈ Z
If x cos θ = `y cos (theta + (2pi)/3) = z cos (theta + (4pi)/3)`. find the value of xy + yz + zx
Prove that sin 4α = `4 tan alpha (1 - tan^2alpha)/(1 + tan^2 alpha)^2`
Prove that `tan (pi/4 + theta) - tan(pi/4 - theta)` = 2 tan 2θ
Express the following as a sum or difference
sin 4x cos 2x
Express the following as a product
cos 65° + cos 15°
Express the following as a product
sin 50° + sin 40°
Show that sin 12° sin 48° sin 54° = `1/8`
Show that `((cos theta -cos 3theta)(sin 8theta + sin 2theta))/((sin 5theta - sin theta) (cos 4theta - cos 6theta))` = 1
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 = `pi/2`, prove the following sin 2A + sin 2B + sin 2C = 4 cos A cos B cos 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:
Let fk(x) = `1/"k" [sin^"k" x + cos^"k" x]` where x ∈ R and k ≥ 1. Then f4(x) − f6(x) =
