Prove that: (45^ +1 2 )= 1+ 1- - Vidyadip

Question

Prove that:
$
\tan \left(45^{\circ}+\frac{1}{2}\right)=\sqrt{\frac{1+\sin \phi}{1-\sin \phi}}
$

✓

Answer

$\begin{aligned}\text { L.H.S. }= & \tan \left(45^{\circ}+\frac{1}{2} \phi\right) \\& =\frac{\tan 45^{\circ}+\tan \frac{\phi}{2}}{1-\tan 45^{\circ} \tan \frac{\phi}{2}} \\& =\frac{1+\tan \phi / 2}{1-\tan \phi / 2}=\frac{1+\frac{\sin \phi / 2}{\cos \phi / 2}}{1-\frac{\sin \phi / 2}{\cos \phi / 2}} \\& =\frac{\cos \phi / 2+\sin \phi / 2}{\cos \phi / 2-\sin \phi / 2} \\& =\frac{(\cos \phi / 2+\sin \phi / 2)(\cos \phi / 2+\sin \phi / 2)}{(\cos \phi / 2-\sin \phi / 2)(\cos \phi / 2+\sin \phi / 2)} \\& =\frac{\cos \phi / 2+\sin ^2 \phi / 2+2 \sin \phi / 2 \cos \phi / 2}{\cos ^2 \phi / 2-\sin \phi / 2} \\& =\frac{1+\sin \phi}{\cos \phi} \\\text { R.H.S. } & =\sqrt{\frac{1+\sin \phi}{1-\sin \phi}}\end{aligned}$
Multiplying the numerator and denominator by
$\begin{aligned}\sqrt{1+\sin \phi} & \\& =\sqrt{\frac{1+\sin \phi}{1-\sin \phi}} \times \sqrt{\frac{1+\sin \phi}{1+\sin\phi}}=\sqrt{\frac{(1+\sin \phi)^2}{1-\sin ^2 \phi}} \\& =\sqrt{\frac{(1+\sin \phi)^2}{\cos ^2 \phi}}=\frac{1+\sin \phi}{\cos \phi}\end{aligned}$
$\text {L.H.S. = R.H.S.}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from PART - 1 CH - 3 Trigonometric Functions→PART - 1 CH - 3 Trigonometric Functions — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

For any two sets A and B, show that the following statements are equevalent:
$\text{A}\subset\text{B}.$

If $\tan\text{x}=\frac{\text{a}}{\text{b}},$ show that $\frac{\text{x}\sin\text{x - b}\cos\text{x}}{\text{a}\sin\text{x}+\text{b}\cos\text{x}}=\frac{\text{a}^2-\text{b}^2}{\text{a}^2+\text{b}^2}.$

Find the length of the perpendicular from the origin to the straight line joining the two points whose coordinates are $(\text{a} \cos \alpha, \text{a} \sin \alpha)$ and $(\text{a} \cos \beta, \text{a} \sin\beta).$

If $|z|=1$, then prove that $\frac{z-1}{z+1},(z \neq-1)$ is a pure imaginary number. If $z=1$, then what conclusion do you draw from this?

Find the values of the other five trigonometric functions in the following:
$\cot\text{x}=-\frac{1}{2},$ x in quadrant II

Find the mean deviation about the median for the data

x_i	5	7	9	10	12	15
f_i	8	6	2	2	2	6

Find the length of the line-segment joining the vertex of the parabola y² = 4ax and a point on the parabola where the line-segment makes an angle $\theta$ to the x-axis.

In a class of 35 students, 24 like to play cricket and 16 like to play football. Also, each student likes to play at least one of the two games. How many students like to play both cricket and football?

A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the path traced by the man.

Find the real values of $\theta$ for which the complex number $\frac{1+\text{i}\cos\theta}{1-2\text{i}\cos\theta}$ is purely real.