If p = cos x - sin x, q = 1-sin^3 x/1-sin x , r = 1+cos^3 x / 1+cos x, what is the value of p + q + r?

Question

If p = cos x - sin x, q = \(\frac{1 - \sin^3 x}{1 - \sin x}\), r = \(\frac{1 + \cos ^3x}{1 + \cos x }\), what is the value of p + q + r?

(a) 0

(b) 1

(c) 2

(d) 3

Answer 1

Correct option is (d) 3

p = cos x − sin x, q = \(\frac{1 - \sin^3 x}{1 - \sin x}\), r = \(\frac{1 + \cos ^3x}{1 + \cos x }\),

\(q = \frac{(1 - \sin x)(1 + \sin^2x + \sin x)}{1 - \sin x}\)      [\(\because\) a³ − b³ = ( a − b) (a² + ab + b²)]

= 1 + sin²x + sin x

\(r = \frac{(1 + \cos x) (1 + \cos^2x - \cos x)}{1 + \cos x}\)      [\(\because\) a³ − b³ = ( a − b) (a² + ab + b²)]

= 1 + cos x − cos x

Now, p + q + r = cos x − sin x + 1 + sin²x + sin x + 1 + cos²x − cos x

= 2 + sin²x + cos²x    [\(\because\) sin²x + cos²x = 1]

= 3