Q. 175.0( 1 Vote )

# If x 0, then 1 + sec x + sec^{2}x + sec^{3}x + sec^{4}x + sec^{5}x is equal to

A. (1 sec x) (sec^{2}x + sec^{3}x + sec^{4}x)

B. (1 + sec x) (1 + sec^{2}x + sec^{4}x)

C. (1 sec x) (sec x + sec^{3}x + sec^{5}x)

D. (1 + sec x) (1 + sec^{3}x + sec^{4}x)

Answer :

We see that the problem under consideration is a GP with a = 1 and common ratio r = sec x.

Sum of GP S =

Where n = number of terms, here n = 6.

Therefore

S =

The numerator is of the form a^{6}-b^{6}

a^{6}-b^{6} = (a + b)(a-b)(a^{2} + b^{2} + ab)(a^{2} + b^{2}-ab)

Therefore (sec x)^{6} -1^{6} = (sec x + 1)(sec x-1)(sec^{2}x + 1 + sec x)(sec^{2}x + 1-sec x)

(sec x)^{6} -1^{6} = (sec x + 1)(sec x-1)(1 + sec^{2}x + sec^{4}x)

S = = (sec x + 1)(1 + sec^{2}x + sec^{4}x)

So the correct answer is option B.

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