Questions: (sin^2 theta)/(cos theta(1+cos theta))+(1+cos theta)/cos theta = (a) cosec theta (b) sec theta (c) 2 cos theta (d) 2 sec theta

Transcript text: \[ \frac{\sin ^{2} \theta}{\cos \theta(1+\cos \theta)}+\frac{1+\cos \theta}{\cos \theta}= \] (a) $\operatorname{cosec} \theta$ (b) $\sec \theta$ (c) $2 \cos \theta$ (d) $2 \sec \theta$

Solution

Solution Steps

Step 1: Rewrite sin²θ using the Pythagorean identity

We know that sin²θ + cos²θ = 1. Therefore, sin²θ = 1 - cos²θ. Substitute this into the given expression:

(1 - cos²θ)/(cosθ(1 + cosθ)) + (1 + cosθ)/cosθ

Step 2: Simplify the first term

We can factor the numerator of the first term as a difference of squares:

1 - cos²θ = (1 - cosθ)(1 + cosθ)

Substitute this into the first term:

(1 - cosθ)(1 + cosθ)/(cosθ(1 + cosθ)) = (1 - cosθ)/cosθ

Step 3: Combine the terms

Now we have:

(1 - cosθ)/cosθ + (1 + cosθ)/cosθ = (1 - cosθ + 1 + cosθ)/cosθ = 2/cosθ

Step 4: Simplify using the reciprocal identity

Recall that secθ = 1/cosθ. Therefore, 2/cosθ can be rewritten as 2secθ.

Final Answer: The correct answer is (d) 2secθ

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