# If x sin^{3}θ + y cos^{3}θ = sin θ cos θ and x sin θ – y cos θ = 0, then prove that x^{2} + y^{2} = 1

Trigonometry is a discipline of mathematics that studies the relationships between the lengths of the sides and angles of a right-angled triangle. Trigonometric functions, also known as goniometric functions, angle functions, or circular functions, are functions that establish the relationship between an angle to the ratio of two of the sides of a right-angled triangle. The six main trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant.

Angles defined by the ratios of trigonometric functions are known as trigonometry angles. Trigonometric angles represent trigonometric functions. The value of the angle can be anywhere between 0-360°.

As given in the above figure in a right-angled triangle:

**Hypotenuse:**The side opposite to the right angle is the hypotenuse, It is the longest side in a right-angled triangle and opposite to the 90° angle.**Base:**The side on which angle C lies is known as the base.**Perpendicular:**It is the side opposite to angle C in consideration.

**Trigonometric Functions**

Trigonometry has 6 basic trigonometric functions, they are sine, cosine, tangent, cosecant, secant, and cotangent. Now let’s look into the trigonometric functions. The six trigonometric functions are as follows,

sine:It is defined as the ratio of perpendicular and hypotenuse and It is represented as sin θ

cosine:It is defined as the ratio of base and hypotenuse and it is represented as cos θ

tangent:It is defined as the ratio of sine and cosine of an angle. Thus the definition of tangent comes out to be the ratio of perpendicular and base and is represented as tan θ

cosecant:It is the reciprocal of sin θ and is represented as cosec θ.

secant:It is the reciprocal of cos θ and is represented as sec θ.

cotangent:It is the reciprocal of tan θ and is represented as cot θ.

**Trigonometric Ratios**

Sin θ = Perpendicular / Hypotenuse = AB/AC

Cosine θ = Base / Hypotenuse = BC/AC

Tangent θ = Perpendicular / Base = AB/BC

Cosecant θ = Hypotenuse / Perpendicular = AC/AB

Secant θ = Hypotenuse / Base = AC/BC

Cotangent θ = Base / Perpendicular = BC/AB

**Reciprocal Identities**

Sin θ = 1/Cosec θ OR Cosec θ = 1/Sin θ

Cos θ = 1/Sec θ OR Sec θ = 1/Cos θ

Cot θ = 1/Tan θ OR Tan θ = 1/Cot θ

Cot θ = Cos θ/Sin θ OR Tan θ = Sin θ/Cos θ

Tan θ.Cot θ = 1

**Values of Trigonometric Ratios**

0° | 30° | 45° | 60° | 90° | |
---|---|---|---|---|---|

Sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |

Cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |

Tan θ | 0 | 1/√3 | 1 | √3 | Not Defined |

cosec θ | Not Defined | 2 | √2 | 2/√3 | 1 |

Sec θ | 1 | 2/√3 | √2 | 2 | Not Defined |

Cot θ | Not Defined | √3 | 1 | 1/√3 | 0 |

Trigonometric Identities of Complementary and Supplementary Angles

**Complementary Angles:**Pair of angles whose sum is equal to 90°**Supplementary Angles:**Pair of angles whose sum is equal to 180°

**Identities of Complementary angles are**

sin (90° – θ) = cos θ

cos (90° – θ) = sin θ

tan (90° – θ) = cot θ

cot (90° – θ) = tan θ

sec (90° – θ) = cosec θ

cosec (90° – θ) = sec θ

**Identities of Supplementary Angles**

sin (180° – θ) = sin θ

cos (180° – θ) = – cos θ

tan (180° – θ) = – tan θ

cot (180° – θ) = – cot θ

sec (180° – θ) = – sec θ

cosec (180° – θ) = – cosec θ

**If x sin**^{3} θ + y cos^{3} θ = sin θ cos θ and x sin θ – y cos θ = 0, then prove that x^{2} + y^{2} = 1, (where, sin θ ≠ 0 and cos θ ≠ 0).

^{3}

**Solution:**

Here we have, x sin

^{3}θ + y cos^{3}θ = sin θ cos θGiven:

x sin

^{3 }θ + y cos^{3 }θ = sin θ cos θ⇒ (x sin θ) sin

^{2}θ + (y cos θ) cos^{2}θ = sin θ cos θ⇒ (x sin θ) sin

^{2}θ + (x sin θ) cos^{2}θ = sin θ cos θ (∵ y cos θ = x sin θ)⇒ x sin θ(sin

^{2}θ + cos^{2}θ) = sin θ cos θ (sin^{2 }θ + cos^{2 }θ = 1)⇒ x sin θ = sin θ cos θ

⇒

x = cos θ ….(eq. 1)Now another trigono eq we have x sin θ – y cos θ = 0

We can write it as x sin θ = y cos θ

from eq. 1 we have x = cos θ so put it in above eq. x sin θ = y cos θ

So x sin θ = y cos θ

cos θ sin θ = y cos θ

y = sin θ eq. 2

Now by squaring and adding both the equation 1 & 2x

=cos θ & y = sin θx

^{2}= cos^{2}θ & y^{ 2}= sin^{2 }θSo now

x

^{2}+ y^{2}= cos^{2}θ + sin^{2}θ { cos^{2}θ + sin^{ 2}θ = 1 }

x^{2}+ y^{2}= 1

Hence proved

**Similar Questions**

**Question 1: If cos θ + sin θ = √2 cos θ, prove that cos θ – sin θ = √2 sin θ?**

**Solution:**

Here we have

cos θ + sin θ = √2 cos θ

Squaring both the sides

(cos θ + sin θ)

^{2}= (√2 cos θ)^{2}cos

^{2}θ + sin^{2}θ + 2 sin θcos θ = 2 cos^{2}θcos

^{2}θ – sin^{2}θ = 2 sin θcos θ(cos θ + sin θ)(cos θ – sin θ) = 2 sin θcos θ

(cos θ – sin θ) = 2 sin θcos θ/(cos θ + sin θ)

cos θ – sin θ = 2 sin θcos θ/√2 cos θ {cos θ + sin θ = √2 cos θ}

cos θ – sin θ = √2 sin θ

Hence proved

**Question 2: Eliminate theta from the equation: tan θ – cot θ = a and cos θ + sin θ = b**

**Solution: **

tan θ – cot θ = a ………. (1)

cos θ + sin θ = b ………. (2)

Squaring both sides of (eq2) we get,

(cos θ + sin θ)

^{2}= (b)^{2}cos

^{2}θ + sin^{2}θ + 2cos θ sin θ = b^{2 }{sin^{2}θ + cos^{2}θ = 1}1 + 2 cos θ sin θ = b

^{2}2 cos θ sin θ = b

^{2 }– 1 ………. (3)Again, from (1)

We get, (sin θ/cos θ) – (cos θ/sin θ) = a

(sin

^{2}θ – cos^{2}θ)/(cos θ sin θ) = asin

^{2}θ – cos^{2}θ = a sin θ cos θ(sin θ + cos θ) (sin θ – cos θ) = a ∙ (b

^{2}– 1)/2 ………. [by (3)]b(sin θ – cos θ) = (½) a (b

^{2}– 1) [by (2)]b

^{2}(sin θ – cos θ)^{2 }= (1/4) a^{2}(b^{2}– 1)2 [Squaring both the sides]b

^{2}[(sin θ + cos θ)^{2}– 4 sinθ cos θ] = (1/4) a^{2}(b^{2}– 1)2b

^{2 }[b^{2}– 2 ∙ (b^{2}– 1)] = (1/4) a^{2}(b^{2}– 1)2 [from (2) and (3)]

4b^{2}(2 – b^{2}) = a^{2}(b^{2}– 1)2which is the required θ-eliminate.