If θ = 30°, verify that: 1 - sin 2θ = (sinθ - cosθ)2 - Mathematics

If θ = 30°, verify that: 1 - sin 2θ = (sinθ - cosθ)²

Sum

Given: θ = 30°
1 - sin2θ
= 1 - sin2 x 30°
= 1 - sin60°

= `1 - sqrt(3)/(2)`

= `(2 - sqrt(3))/(2)`
(sinθ - cosθ)²
= sin²θ + cos²θ - 2sinθ cosθ
= 1 - 2 x sin30° x cos30

= `1 - 2 xx (1)/(2) xx sqrt(3)/(2)`

= `1 - sqrt(3)/(2)`

= `(2 - sqrt(3))/(2)`
⇒ 1 - sin2θ = (sinθ - cosθ)².

shaalaa.com

Trigonometric Equation Problem and Solution

Is there an error in this question or solution?

Chapter 27: Trigonometrical Ratios of Standard Angles - Exercise 27.1

Chapter 27 Trigonometrical Ratios of Standard Angles
Exercise 27.1 | Q 12.5

State for any acute angle θ whether sin θ increases or decreases as θ increases

Solve the following equation for A, if sin 3 A = `sqrt3 /2`

If 2 cos 2A = `sqrt3` and A is acute,
find:
(i) A
(ii) sin 3A
(iii) sin² (75° - A) + cos² (45° +A)

If θ = 30°, verify that: sin2θ = `(2tanθ)/(1 ++ tan^2θ)`

Find x and y, in each of the following figure:

In the given figure; ∠B = 90°, ∠ADB = 30°, ∠ACB = 45° and AB = 24 m. Find the length of CD.

Evaluate the following: `(2sin28°)/(cos62°) + (3cot49°)/(tan41°)`

Evaluate the following: `(5sec68°)/("cosec"22°) + (3sin52° sec38°)/(cot51° cot39°)`

Evaluate the following: `(3sin37°)/(cos53°) - (5"cosec"39°)/(sec51°) + (4tan23° tan37° tan67° tan53°)/(cos17° cos67° "cosec"73° "cosec"23°)`

If sec2θ = cosec3θ, find the value of θ if it is known that both 2θ and 3θ are acute angles.