NCERT Solutions for Chapter: Introduction to Trigonometry, Exercise 1: Exercise

During a maths lesson, Mr. Kumar wrote the expression given below on the board and asked the students to simplify it.

$\frac{\cos \mathrm{A}}{1-\sin \mathrm{A}}+\frac{1-\sin \mathrm{A}}{\cos {\displaystyle }{\displaystyle \mathrm{A}}}$

Salma solved it in her notebook as follows:

$\frac{\cos \mathrm{A}}{1-\sin \mathrm{A}}+\frac{1-\sin \mathrm{A}}{\cos {\displaystyle }{\displaystyle \mathrm{A}}}$

$=\frac{{\displaystyle {\cos }^2\mathrm{A}+(1-\sin \mathrm{A}{)}^2}}{(1-\sin \mathrm{A})\times \cos \mathrm{A}}$   ....(step 1)

$=\frac{{\cos }^{2 }\mathrm{A}+{\cos }^2 \mathrm{A}}{(1-\sin \mathrm{A})\times \cos \mathrm{A}}$      ....(step 2)

$=\frac{2{\cos }^2 \mathrm{A}}{(1-\sin \mathrm{A})\times \cos \mathrm{A}}$      ....(step 3)

$=\frac{2\cos \mathrm{A}}{1-\sin \mathrm{A}}$                   ....(step 4)

Examine if Salma has made any error(s) and rectify them to find the correct answer.

If $\frac{1}{\sin \theta -\cos \theta }=\frac{\cos {\displaystyle e}{\displaystyle c}{\displaystyle \theta }}{\sqrt{2}}$, prove that $(\frac{1}{\sin \theta +\cos \theta }{)}^2=\frac{s{\displaystyle e}{\displaystyle c^2}{\displaystyle \theta }}{2}$.

Prove that: $\frac{\cos e{c}^{2}x-{\sin }^{2}xco{t}^{2}x-co{t}^{2}x}{{\sin }^{2}x}=1$

$∆\mathrm{PQR}$ is inscribed in a circle with a centre $\mathrm{O}$ and radius $r$ units

If $\mathrm{PR}$ is the diameter of the circle and $\angle \mathrm{RQO}=\mathrm{\beta }$ 

Express $({\mathrm{QR}}^4-{\mathrm{PQ}}^4)$, in terms of $r$ and $\mathrm{\beta }$, to the simplest form. 

Show your steps and give valid reasons.

$ABC$ is an isosceles triangle, with $AB=BC.$ A semicircle of an area equal to that of the triangle is combined with it.

What is the value of $\tan x?$

What is the value of $\left(1-\sin \right)\left(\sec +\tan \right)?$

Let $A\left(-1,0,0\right), B\left(0,1,0\right), C\left(0,0,-1\right)$ are the vertices of $△ABC.$ If $R$ and $r$ denote the circumradius and inradius of $△ABC,$ then which of the following statement(s) is/are correct$?$

If $3\sec A-2\cos B=\sqrt{3}$ and $B={30}^{\circ },$ then $\cos \left(A-B\right)$ is equal to