Question

(1 + sin²x)(1 + sin⁴x)(1 + sin⁸x) + sec²xsin¹⁶x is equal to – 

1. sec2x
2. cos²x
3. sin²x
4. cosec²x

Answer 1

Correct Answer - Option 1 : sec2x

Given:

(1 + sin²x)(1 + sin⁴x)(1 + sin⁸x) + sec²xsin¹⁶x

Formula Used:

(1 – sin²x) = cos²x

(A² – B²) = (A + B)(A – B)

Calculation:

(1 + sin²x)(1 + sin⁴x)(1 + sin⁸x) + sec²xsin¹⁶x

Multiply denominator and numerator by (1 – sin²x), we have –

⇒ (1 – sin²x) (1 + sin²x) (1 + sin⁴x) (1 + sin⁸x)/ (1 – sin²x) + sec²xsin¹⁶x

⇒ (1 – sin⁴x) (1 + sin⁴x) (1 + sin⁸x)/ (cos²x) + sec²xsin¹⁶x

⇒ (1 – sin⁸x) (1 + sin⁸x) sec²x + sec²xsin¹⁶x

⇒ (1 – sin¹⁶x) sec²x + sec²xsin¹⁶x

⇒ sec2x - sec²xsin¹⁶x + sec²xsin¹⁶x

⇒ sec2x

∴ The correct answer is sec2x