If *A* = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], then insert the appropriate symbol ∈ or ∉ in each of the following blank space:

12 ...... A

Concept: Sets and Their Representations

Answer the following:

If sinθ = `(x^2 - y^2)/(x^2 + y^2)` then find the values of cosθ, tanθ in terms of x and y.

Concept: Introduction of Trigonometry

Prove the following:

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

Concept: Trigonometric Functions of Sum and Difference of Angles

Prove the following:

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

Concept: Trigonometric Functions of Sum and Difference of Angles

Prove the following:

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

Concept: Trigonometric Functions of Sum and Difference of Angles

Prove the following:

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

Concept: Trigonometric Functions of Sum and Difference of Angles

Prove the following:

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

Concept: Trigonometric Functions of Sum and Difference of Angles

Prove the following:

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

Concept: Trigonometric Functions of Sum and Difference of Angles

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

Concept: Trigonometric Functions of Sum and Difference of Angles

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

Concept: Trigonometric Functions of Sum and Difference of Angles

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

Concept: Trigonometric Functions of Sum and Difference of Angles

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

Concept: Trigonometric Functions of Sum and Difference of Angles

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

Concept: Trigonometric Functions of Sum and Difference of Angles

Select the correct option from the given alternatives :

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

Concept: Trigonometric Functions of Sum and Difference of Angles

Select the correct option from the given alternatives :

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

Concept: Trigonometric Functions of Sum and Difference of Angles

Select the correct option from the given alternatives :

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

Concept: Trigonometric Functions of Sum and Difference of Angles

Prove the following:

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

Concept: Trigonometric Functions of Sum and Difference of Angles

Prove the following:

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

Concept: Trigonometric Functions of Sum and Difference of Angles

Prove the following:

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

Concept: Trigonometric Functions of Sum and Difference of Angles

Prove the following:

3tan^{6}10° – 27 tan^{4}10° + 33tan^{2}10° = 1

Concept: Trigonometric Functions of Sum and Difference of Angles