Question

Evaluate $\frac{1+tan\theta }{1+cot\theta }$, if sin θ = x and cos θ = y.

• A. $\frac{x}{y}$
• B. $\frac{y}{x}$
• C. xy
• D. $\frac{2x}{y}$

Answer 1

The correct option is A

$\frac{x}{y}$

Here, $sin\theta =x$ and $cos\theta =y$.

We can write,

1 + tan θ = $1+\frac{sin\theta }{cos\theta }$ = $\frac{cos\theta +sin\theta }{cos\theta }$

1+ cot θ = $1+\frac{cos\theta }{sin\theta }$ = $\frac{sin\theta +cos\theta }{sin\theta }$

$\Rightarrow \frac{1+tan\theta }{1+cot\theta }$ = $\frac{\frac{cos\theta +sin\theta }{cos\theta }}{\frac{sin\theta +cos\theta }{sin\theta }}=\frac{sin\theta }{cos\theta }=\frac{x}{y}$

$\therefore \frac{1+tan\theta }{1+cot\theta }$ = $\frac{x}{y}$