Question

hi trigonometry PLZ help
it's from RD Sharma
pg 10.59
MCQ set


☞no spamming PLZ it's important​

Answer 1

$\huge \mathsf \orange{ \underline{ \underline\red{Answer}}} \green\downarrow$

Option (b) 0 is correct

$\mathsf \blue{ \underline{ \underline\pink{Step-by-step \: explanation}}} \purple\downarrow$

The expression given in the question is:

$\bf \red{2 \sec^{2} \theta \: + \: 2 \tan^{2}\theta \: - \: 7}$

Now, we are going to convert the above

expression in terms of $\sec^{2} \theta$ using the trigonometric identity:

$\bf \red{1 \: + \: \tan^{2} \theta \: = \: \sec^{2} \theta}$

$\bf \red {\Longrightarrow \: \tan^{2} \theta \: = \: \sec^{2} \theta \: - \: 1}$

Substituting the above value of $\tan^{2} \theta$ in the given expression $4 \sec^{2} \theta \: - \: 9$ we get,

$\bf \red{4( \frac{3}{2})^{2} \: - \: 9}$

$\bf \red{= \: 4( \frac{9}{4})\: - \: 9}$

$\bf \red{= \: 9 \: - \: 9 \: = \: 0}$

From the above calculation, the value of the given expression is 0.

Hence, the correct option is (b).

Note: In the above solution instead of using the identity $1 \: + \: \tan^{2} \theta \: = \: \sec^{2} \theta$ we can substitute the value of the tan θ in the given expression $2\sec^{2}\theta \: + \: \tan^{2} \theta \: - \: 7$. Now, we can find the value of tan θ as follows:

$\bf \red{ \cos \theta \: = \: \frac{2}{3} }$

In the below diagram, we have shown a triangle ABC right angled at B along with an angle $\theta$.

In the above figure, “P” stands for perpendicular with respect to angle θ, “B” stands for the base with respect to angle θ and “H” stands for the hypotenuse with respect to angle $\theta$.

We know that, cosθ is equal to base divided by hypotenuse so using Pythagoras theorem we can find the perpendicular of the triangle.

$\bf \red{ \cos \theta \: = \: \frac{B}{H} }$

$\bf \red{{H^{2} \: = B^{2} \: + \: P^{2} }}$

$\bf \red{\Rightarrow \: 9 \: = \: 4 \: + \: P^{2} }$

$\bf \red{\Rightarrow P^{2} \: = \: 5}$

$\bf \red{\Rightarrow \: P \: = \: \sqrt{5} }$

From the above calculation, we can find the value of tanθ :

$\bf \red{\tan \theta \: = \: \frac{ \sqrt{5} }{2} }$

Substituting this value of tanθ in the given expression we get,

$\bf \red{2 (\sec^{2} \theta \: + \: \tan^{2} \theta) \: - \: 7}$

$\bf \red{= \: 2( \frac{9}{4} \: + \: \frac{5}{4} ) \: - \: 7}$

$\bf \red{= \: 2( \frac{14}{4} ) \: - \: 7}$

$\bf \red{= \: 7 \: - \: 7 \: = \: 0}$

Hence, we have got the answer as 0.

I hope it's helps you ❤️.

Please markerd as brainliest answer.