Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) [English] 11 Standard Maharashtra State Board chapter 3 - Trigonometry - 2 [Latest edition]

Find the value of:

sin 15°

Find the values of:

cos 75°

Find the values of:

tan 105°

Find the values of:

cot 225°

Prove the following:

`cos (pi/2 - x) cos(pi/2 - y) - sin(pi/2 - x) sin(pi/2 - y)` = – cos (x + y)

Prove the following:

`tan(pi/4 + theta) = (1 + tan theta)/(1 - tan theta)`

Prove the following:

`((1 + tan x)/(1 - tan x))^2 = tan(pi/4 + x)/(tan(pi/4 - x))`

Prove the following:

sin [(n + 1)A]. sin [(n + 2)A] + cos [(n + 1)A]. cos [(n + 2)A] = cos A

Prove the following:

`sqrt(2)cos (pi/4 - "A")` = cos A + sin A

Prove the following:

`(cos(x - y))/(cos(x + y)) = (cotx coty + 1)/(cotx coty - 1)`

Prove the following:

cos(x + y).cos(x − y) = cos²y − sin²x

Prove the following:

`(tan5"A" - tan3"A")/(tan5"A" + tan3"A") = (sin2"A")/(sin8"A")`

Prove the following:

tan8θ − tan5θ − tan3θ = tan8θ · tan5θ · tan3θ

Prove the following:

tan50° = tan40° + 2 tan10°

Prove the following:

`(cos27^circ + sin27^circ)/(cos27^circ - sin27^circ)` = tan72°

Prove the following:

tan10° + tan35° + tan10°.tan35° = 1

Prove the following:

`(cot"A"cot4"A" + 1)/(cot"A" cot4"A" - 1) = (cos3"A")/(cos5"A")`

Prove the following:

`(cos15^circ - sin15^circ)/(cos15^circ + sin15^circ) = 1/sqrt(3)`

If sin A = `(-5)/13, pi < "A" < (3pi)/2` and cos B = `3/5, (3pi)/2 < "B" < 2pi` find sin (A + B)

If sin A = `(-5)/13, pi < "A" < (3pi)/2` and cos B = `3/5, (3pi)/2 < "B" < 2pi` find cos (A – B)

If sin A = `(-5)/13, pi < "A" < (3pi)/2` and cos B = `3/5, (3pi)/2 < "B" < 2pi` find tan (A + B)

If tan A = `5/6, tan "B" = 1/11`, prove that A + B = `pi/4`

Find the value of :

sin 690°

Find the value of :

sin (495°)

Find the value of :

cos 315°

Find the value of :

cos (600°)

Find the value of :

tan 225°

Find the value of :

tan (– 690°)

Find the value of :

sec 240°

Find the value of :

sec (– 855°) 

Find the value of :

cosec 780°

Find the value of :

cot (– 1110°)

Prove the following:

`(cos(pi + x) cos(-x))/(sin(pi - x)cos(pi/2 + x))` = cot²x

Prove the following:

`cos((3pi)/2 + x) cos(2pi + x)[cot((3pi)/2 - x) + cot(2pi + x)]` = 1

Prove the following:

sec 840° . cot (– 945°) + sin 600° tan (– 690°) = `3/2`

Prove the following:

`("cosec"(90^circ - x)sin(180^circ - x)cot(360^circ - x))/(sec(180^circ + x)tan(90^circ + x)sin(-x))` = 1

Prove the following:

`(sin^3(pi + x)sec^2(pi - x)tan(2pi - x))/(cos^2(pi/2 + x)sin(pi - x)"cosec"^2 - x)` = tan³x

Prove the following:

cosθ + sin (270° + θ) − sin (270° − θ) + cos (180° + θ) = 0

Find the value of :

`sin  pi/8`

Find the value of :

`cos  pi/8`

Find sin 2x, cos 2x, tan 2x if secx = `(-13)/5, pi/2 < x < pi`

Prove the following:

`(1 - cos2theta)/(1 + cos2theta)` = tan²θ

Prove the following:

(sin 3x + sin x)sin x + (cos 3x – cos x) cos x = 0

Prove the following:

(cos x + cos y)² + (sin x – sin y)² = `4cos^2  ((x + y))/2`

Prove the following:

(cos x – cos y)² + (sin x – sin y)² = `4sin^2  ((x - y))/2`

Prove the following:

tan x + cot x = 2 cosec 2x

Prove the following:

`(cosx + sinx)/(cosx - sinx) - (cosx - sinx)/(cosx + sinx)` = 2tan2x

Prove the following:

`sqrt(2 + sqrt(2 + sqrt(2 + 2cos8x)` = 2 cos x

Prove the following:

16 sin θ cos θ cos 2θ cos 4θ cos 8θ = sin 16θ

Prove the following:

`(sin3x)/cosx + (cos3x)/sinx` = 2 cot 2x

Prove the following:

`cosx/(1 + sinx) = (cot(x/2) - 1)/(cot(x/2) + 1)`

Prove the following:

`(tan(theta/2) + cot(theta/2))/(cot(theta/2) - tan(theta/2))` = secθ

Prove the following:

`1/(tan3"A" - tan"A") - 1/(cot3"A" - cot"A")` = cot2A

Prove the following:

cos7° cos 14° cos28° cos 56° = `sin68^circ/(16cos83^circ)`

Prove the following:

`(sin^2(-160^circ))/(sin^(2)70^circ) + sin(180^circ - theta)/sintheta` = sec²20°

Prove the following:

`(2cos 4x + 1)/(2cosx + 1)` = (2 cos x – 1) (2 cos 2x – 1)

Prove the following:

cos²x + cos²(x + 120°) + cos²(x – 120°) = `3/2`

Prove the following:

2cosec 2x + cosec x = `secx cot(x/2)`

Prove the following:

`4 cos x. cos(x + pi/3) . cos (x - pi/3)` = cos 3x

Prove the following:

`sinx tan(x/2) + 2cosx = 2/(1 + tan^2(x/2))`

Express the following as a sum or difference of two trigonometric function:

2 sin 4x cos 2x

Express the following as a sum or difference of two trigonometric function:

`2sin  (2pi)/3 cos  pi/2`

Express the following as a sum or difference of two trigonometric function:

2 cos 4θ cos 2θ

Express the following as a sum or difference of two trigonometric function:

2 cos 35° cos 75°

Prove the following :

`(sin2x + sin2y)/(sin2x - sin2y) = (tan(x + y))/(tan(x - y))`

Prove the following :

sin 6x + sin 4x – sin 2x = 4 cos x sin 2x cos 3x

Prove the following :

`(sinx - sin3x + sin5x - sin7x)/(cosx - cos3x - cos5x + cos7x)` = cot2x

Prove the following :

sin 18° cos 39° + sin 6° cos 15° = sin 24° cos 33°

Prove the following :

cos 20° cos 40° cos 60° cos 80° = `1/16`

Prove the following :

sin 20° sin 40° sin 60° sin 80° = `3/16`

In ΔABC, A + B + C = π show that 

cos 2A + cos 2B + cos 2C = –1 – 4 cos A cos B cos C

In ΔABC, A + B + C = π show that

sin A + sin B + sin C = `4cos  "A"/2  cos  "B"/2  cos  "C"/2 `

In ΔABC, A + B + C = π show that

cos A + cos B – cos C = `4cos  "A"/2  cos  "B"/2  sin  "C"/2 - 1`

In ΔABC, A + B + C = π show that

sin²A + sin²B − sin²C = 2 sin A sin B cos C

In ΔABC, A + B + C = π show that

`sin^2  "A"/2 + sin^2  "B"/2 - sin^2  "C"/2 = 1 - 2cos  "A"/2  cos  "B"/2 sin  "C"/2`

In ΔABC, A + B + C = π show that

`cot  "A"/2 + cot  "B"/2 + cot  "C"/2 = cot  "A"/2  cot  "B"/2 cot  "C"/2`

In ΔABC, A + B + C = π show that

tan 2A + tan 2B + tan 2C = tan 2A tan 2B tan 2C

In ΔABC, A + B + C = π show that

cos²A +cos²B – cos²C = 1 – 2 sin A sin B cos C

Select the correct option from the given alternatives :

The value of sin(n + 1) A sin (n + 2) A + cos(n + 1) A cos(n + 2) A is equal to

Select the correct option from the given alternatives :

If tan A – tan B = x and cot B – cot A = y, then cot (A – B) = _____

Select the correct option from the given alternatives :

If sin θ = n sin (θ + 2α), then tan (θ + α) is equal to

Select the correct option from the given alternatives:

The value of `costheta/(1 + sin theta)` is equal to .....

Select the correct option from the given alternatives :

The value of cos A cos (60° – A) cos (60° + A) is equal to ......

Select the correct option from the given alternatives :

The value of `sin  pi/14sin  (3pi)/14sin  (5pi)/14sin  (7pi)/14sin  (9pi)/14sin  (11pi)/14sin  (13pi)/14` is ....

Select the correct option from the given alternatives :

If α + β + γ = π then the value of sin²α + sin²β – sin²γ is equal to …......

Select the correct option from the given alternatives :

Let 0 < A, B < `pi/2` satisfying the equation 3 sin²A + 2 sin²B = 1 and 3 sin 2A − 2 sin 2B = 0 then A + 2B is equal to ______

Select the correct option from the given alternatives :

In ∆ABC if cot A cot B cot C > 0 then the triangle is _________

Select the correct option from the given alternatives :

The numerical value of tan 20° tan 80° cot 50° is equal to ______.

Prove the following:

tan 20° tan 80° cot 50° = `sqrt(3)`

Prove the following:

If sin α sin β − cos α cos β + 1 = 0 then prove cot α tan β = −1

Prove the following:

`cos  (2pi)/15 cos  (4pi)/15cos  (8pi)/15cos  (16pi)/15 = 1/16`

Prove the following:

`(1 + cos  pi/8)(1 + cos  (3pi)/8)(1 + cos  (5pi)/8)(1 + cos  (7pi)/8) = 1/8`

Prove the following:

cos 12°+ cos 84° + cos 156° + cos 132° = `-1/2`

Prove the following:

`cos(pi/4 + x) + cos(pi/4 - x) = sqrt(2)cosx`

Prove the following:

`(sin5x - 2sin3x + sinx)/(cos5x - cosx)` = tanx

Prove the following:

sin²6x − sin²4x = sin2x sin10x

Prove the following:

cos²2x − cos²6x = sin4x sin8x

Prove the following:

cot4x (sin5x + sin3x) = cotx (sin5x − sin3x)

Prove the following:

`(cos9x - cos5x)/(sin17x - sin3x) = - (sin2x)/(cos10x)`

Prove the following:

If sin 2A = λsin 2B then prove that `(tan("A" + "B"))/(tan("A" - "B")) = (lambda + 1)/(lambda - 1)`

Prove the following:

`(2cos2"A" + 1)/(2cos2"A" - 1)` = tan(60° + A) tan(60° − A)

Prove the following:

tanA + tan(60° + A) + tan(120° + A) = 3 tan 3A

Prove the following:

3tan⁶10° – 27 tan⁴10° + 33tan²10° = 1

Prove the following:

cosec 48° + cosec 96° + cosec 192° + cosec 384° = 0

Prove the following:

3(sin x – cos x)⁴ + 6(sin x + cos x)² + 4(sin⁶x + cos⁶x) = 13

Prove the following:

tan A + 2 tan 2A + 4 tan 4A + 8 cot 8A = cot A

Prove the following:

If A + B + C = `(3pi)/2`, then cos 2A + cos 2B + cos 2C = 1 − 4 sin A sin B sin C

Prove the following:

In any triangle ABC, sin A − cos B = cos C then ∠B = `pi/2`.

Prove the following:

`tan^3x/(1 + tan^2x) + cot^3x/(1 + cot^2x)` = secx cosecx − 2sinx cosx 

Prove the following:

sin 20° sin 40° sin 80° = `sqrt(3)/8`

Prove the following:

sin 18° = `(sqrt(5) - 1)/4`

Prove the following:

cos 36° = `(sqrt(5) + 1)/4`

Prove the following:

sin 36° = `(sqrt(10 - 2sqrt(5)))/4`

Prove the following:

`sin  pi^"c"/8 = 1/2sqrt(2 - sqrt(2))`

Prove the following:

`tan  pi/8 = sqrt(2) - 1`

Prove the following:

tan6° tan42° tan66° tan78° = 1

Prove the following:

sin47° + sin61° − sin11° − sin25° = cos7°

Prove the following:

`sqrt(3)  "cosec"20^circ - sec20^circ` = 4

Prove the following:

In ∆ABC, ∠C = `(2pi)/3`, then prove that cos²A + cos²B − cos A cos B = `3/4`