What is the best solution of introduction of trigonometry of class 10 ncert please help me solve this question no. 4(2) of exercise 8.4 give me the best solution of this optional question.Choose the correct option. Justify your choice.(ii) (1 + tan θ + sec θ) (1 + cot θ – cosec θ) (A) 0 (B) 1 (C) 2 (D) – 1 .

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# Choose the correct option. Justify your choice.(ii) (1 + tan θ + sec θ) (1 + cot θ – cosec θ) (A) 0 (B) 1 (C) 2 (D) – 1 . Q.4(2)

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(C) is correct

Justification:

(1 + tan θ + sec θ) (1 + cot θ – cosec θ)

We know that, tan θ = sin θ/cos θ

sec θ = 1/ cos θ

cot θ = cos θ/sin θ

cosec θ = 1/sin θ

Now, substitute the above values in the given problem, we get

= (1 + sin θ/cos θ + 1/ cos θ) (1 + cos θ/sin θ – 1/sin θ)

Simplify the above equation,

= (cos θ +sin θ+1)/cos θ × (sin θ+cos θ-1)/sin θ

= (cos θ+sin θ)

^{2}-1^{2}/(cos θ sin θ)= (cos

^{2}θ + sin^{2}θ + 2cos θ sin θ -1)/(cos θ sin θ)= (1+ 2cos θ sin θ -1)/(cos θ sin θ) (Since cos

^{2}θ + sin^{2}θ = 1)= (2cos θ sin θ)/(cos θ sin θ) = 2

Therefore, (1 + tan θ + sec θ) (1 + cot θ – cosec θ) =2