Question

CET question...
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Answer 1

Answer:

Step-by-step explanation:

Answer 2

What is trigonometry ?

 Trigonometry is the study of angles and the ratios of sine, cosine, tangent, cosec, sectant and cot. It includes a number of trigonometric identity laws and some relation between ratios.

Some of the laws are :

   • sinθ * cosecθ = 1

   • cosθ * secθ = 1

   • tanθ * cotθ = 1

   • sin(90° - θ) = cosθ and vice versa

   • tan(90° - θ) = secθ and vice versa

   • cot(90° - θ) = cosecθ and vice versa

   • sin²θ + cos²θ = 1

   • sec²θ - tan²θ = 1

   • cosec²θ - cot²θ = 1

Solution :

Now, $\frac{tan\theta}{sec\theta + 1} + \frac{sec\theta + 1}{tan\theta}$

= $\frac{tan\theta*tan\theta+(sec\theta+1)(sec\theta+1)}{(sec\theta+1)tan\theta}$

= $\frac{tan^{2}\theta+sec^{2}\theta+2sec\theta+1}{(sec\theta+1)tan\theta}$

= $\frac{(tan^{2}\theta+1)+sec^{2}\theta+2sec\theta}{(sec\theta+1)tan\theta}$

= $\frac{sec^{2}\theta+sec^{2}\theta+2sec\theta}{(sec\theta+1)tan\theta}$

{ since $sec^{2}\theta-tan^{2}\theta=1$ }

= $\frac{2sec^{2}+2sec\theta}{(sec\theta+1)tan\theta}$

= $\frac{2sec\theta(sec\theta+1)}{(sec\theta+1)tan\theta}$

= $\frac{2sec\theta}{tan\theta}$

{by cancelling $(sec\theta+1)$ from both the numerator and the denominator}

= $\frac{\frac{2}{cos\theta}}{\frac{sin\theta}{cos\theta}}$

= $\frac{2cos\theta}{sin\theta*cos\theta}$

= $\frac{2}{sin\theta}$

= 2 cosecθ

Thus, option ( 1 ) is correct.