   Question Answers

# Prove the following; $(1{\text{ + cot A - cosec A)(1 + tan A + sec A) = 2}}$  Hint: In order to solve this question easily we will transform the given terms in of sin and cos. In this question we have to prove that the left - hand side is equal to the right - hand side.

Now, by using trigonometric identities we will easily solve the given problem. We know that $\sin {\text{A = }}\dfrac{1}{{\cos ec{\text{A}}}}$, ${\text{cosA = }}\dfrac{1}{{sec{\text{A}}}}$, ${\text{cotA = }}\dfrac{{\cos {\text{A}}}}{{\sin {\text{A}}}}$, ${\text{tanA = }}\dfrac{{\sin {\text{A}}}}{{\cos {\text{A}}}}$. By using these properties, we will solve this question. Now, putting these values in the given question, we get
L. H. S = $\left( {1 + {\text{ }}\dfrac{{\cos {\text{A}}}}{{\sin {\text{A}}}}{\text{ - }}\dfrac{1}{{\sin {\text{A}}}}} \right)\left( {1{\text{ + }}\dfrac{{\sin {\text{A}}}}{{\cos {\text{A}}}}{\text{ + }}\dfrac{1}{{\cos {\text{A}}}}} \right)$
$\Rightarrow$ L. H. S = $\left( {\dfrac{{\sin {\text{A + cosA - 1}}}}{{\sin {\text{A}}}}} \right)\left( {\dfrac{{\cos {\text{A + sinA + 1}}}}{{\cos {\text{A}}}}} \right)$
Now we can see that in the above equation we can use the property ${{\text{a}}^2} - {{\text{b}}^2} = ({\text{a - b)(a + b)}}$
So, applying this property we get
$\Rightarrow$ L. H. S = $\left( {\dfrac{{{{(\sin {\text{A + cosA)}}}^2}{\text{ }} - {\text{ }}1}}{{\sin {\text{A cosA}}}}} \right)$
By solving further, we get
$\Rightarrow$ L. H. S = $\left( {\dfrac{{({{\sin }^2}{\text{A + co}}{{\text{s}}^2}{\text{A + 2cosA sinA) }} - {\text{ }}1}}{{\sin {\text{A cosA}}}}} \right)$
$\Rightarrow$ L. H. S = $\left( {\dfrac{{2\sin {\text{A cosA}}}}{{\sin {\text{A cosA}}}}} \right)$ as ${\sin ^2}{\text{A + }}{\text{co}}{{\text{s}}^2}{\text{A = 1}}$
$\Rightarrow$ L. H. S = 2 = R. H. S
Hence proved.

Note: To solve questions which include trigonometric terms it is suggested that you should simplify the given term by converting it into sin or cos whichever is possible. Converting in sin or cos simplify the term and you can easily solve the given term. Use identity ${\sin ^2}{\text{A + }}{\text{co}}{{\text{s}}^2}{\text{A = 1}}$ properly after converting the trigonometric term.
View Notes
The Difference Between an Animal that is A Regulator and One that is A Conformer  Relation Between the Length of a Given Wire and Tension for Constant Frequency Using Sonometer  The Making of a Scientist  Changing the Period of a Pendulum  Tan 0 Degrees  Area of a Sector of a Circle Formula  Where There is a Will There is a Way Essay  NCERT Books Class 6 English a Pact with the Sun  CBSE Class 11 Maths Formulas  Area of a Rhombus Formula  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 8 - A Pact with the Sun  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 1 - A Tale of Two Birds  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 10 - A Strange Wrestling Match  Important Questions for CBSE Class 11 Physics Chapter 4 - Motion in a Plane  Important Questions for CBSE Class 6 English A Pact with The Sun  Important Questions for CBSE Class 11 Business Studies Chapter 7 - Formation of a Company  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 2 - The Friendly Mongoose  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 3 - The Shepherd's Treasure  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 6 - The Monkey and the Crocodile  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 7 - The Wonder Called Sheep  CBSE Class 10 Hindi A Question Paper 2020  Hindi A Class 10 CBSE Question Paper 2009  Hindi A Class 10 CBSE Question Paper 2015  Hindi A Class 10 CBSE Question Paper 2016  Hindi A Class 10 CBSE Question Paper 2012  Hindi A Class 10 CBSE Question Paper 2010  Hindi A Class 10 CBSE Question Paper 2007  Hindi A Class 10 CBSE Question Paper 2013  Hindi A Class 10 CBSE Question Paper 2008  Hindi A Class 10 CBSE Question Paper 2014  RS Aggarwal Class 11 Solutions Chapter-16 Conditional Identities Involving the Angles of a Triangle  RD Sharma Class 11 Solutions Chapter 1 - Sets (Ex 1.1) Exercise 1.1  RS Aggarwal Class 8 Mathematics Solutions for Chapter-18 Area of a Trapezium and a Polygon  RS Aggarwal Class 10 Solutions - Trigonometric Ratios  NCERT Solutions for Class 12 English Kaliedoscope Poetry Chapter 1 A Lecture Upon The Shadow  NCERT Solutions for Class 11 Maths Chapter 1 Sets (Ex 1.1) Exercise 1.1  NCERT Solutions Class 11 English Woven Words Poem Chapter 11 Ode to a Nightingale  RD Sharma Solutions for Class 10 Maths Chapter 5 - Trigonometric Ratios  NCERT Solutions for Class 12 English Kaliedoscope Poetry Chapter 4 Kubla Khan or a Vision in a Dream a Fragment  NCERT Solutions for Class 6 English A pact with the Sun  