Question

(1 + tan θ + sec θ) (1 + cot θ − cosec θ) = ______.

Options

• 0

• 1

• 2

• -1

• none of these

Answer 1

(1 + tan θ + sec θ) (1 + cot θ − cosec θ) = 2.

Explanation:

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

= `(1+ (sin theta)/(cos theta)+1/(costheta))(1+(costheta)/(sin theta)-1/(sin theta))`

= `((costheta+sintheta +1)/costheta)((sintheta+cos theta -1)/sintheta)`

= `((sintheta+costheta)^2-(1)^2)/(sinthetacostheta)`

= `(sin^2theta+cos^2 theta + 2sin theta cos theta -1)/(sinthetacostheta)`

= `(1+2sinthetacostheta -1)/(sinthetacostheta)`

= `(2sintheta costheta)/(sin theta costheta)`

= 2

Hence, alternative 2 is correct.