The value of sin²30 – sin ² 40 + sin²  45 – sin² 55 – sin² 35 + sin² 45 – sin ²50 + sin² 60 is:

Correct option is D

Given:

sin²30 – sin ² 40 + sin² 45 – sin² 55 – sin² 35 + sin² 45 – sin ²50 + sin² 60

Formula Used:

$$\sin^2(\theta) + \cos^2(\theta) = 1 \\$$​

​$$\sin(90^\circ - \theta) = \cos(\theta) \\$$​

​$$\sin^2(30^\circ) = \frac{1}{4}, \quad \sin^2(45^\circ) = \frac{1}{2}, \quad \sin^2(60^\circ) = \frac{3}{4} \\$$​​

Solution:

​$$\text{Substitute these into the expression:} \\$$​​

​$$\frac{1}{4} - \sin^2(40^\circ) + \frac{1}{2} - \sin^2(55^\circ) - \cos^2(90^\circ -35^\circ) + \frac{1}{2} - \cos^2(90^\circ -50^\circ) + \frac{3}{4} \\$$​​

​$$\frac{1}{4} - (\sin^2(40^\circ)+ \cos^2(40^\circ)) + \frac{1}{2} - (\sin^2(55^\circ) + \cos(55^\circ)) + \frac{1}{2} - + \frac{3}{4} \\$$​​

​$$\frac{1}{4} + \frac{3}{4} +\frac{1}{2} + \frac{1}{2} - (1) - (1) \\$$​​

​$$(1) + (1) - (1) - (1) \\$$​​

= 0