Answer
375.1k+ views
Hint: Here, use the trigonometric functions and its simplification, then substitute the given cosine angle value and simplify using basic mathematical operations and different properties of the difference of the squares and square-roots. That is $ n = \sqrt n \times \sqrt n $ .
Complete step-by-step answer:
$ b\cos \theta = a $
Make $ \cos \theta $ the subject –
$ \cos \theta = \dfrac{a}{b}\;{\text{ }}.......{\text{(1)}} $
Using the trigonometric identity that –
$ {\sin ^2}\theta + {\cos ^2}\theta = 1 $
Make the subject
$
\Rightarrow {\sin ^2}\theta = 1 - {\cos ^2}\theta \\
$ \Rightarrow $ \sin \theta = \sqrt {1 - {{\cos }^2}\theta } \\
$
Place the given value in the right hand side of the equation –
$ \Rightarrow \sin \theta = \sqrt {1 - {{\left( {\dfrac{a}{b}} \right)}^2}} $
Simplify the above right hand side of the equation –
\[\Rightarrow \sin \theta = \sqrt {1 - {{\dfrac{a}{{{b^2}}}}^2}} \]
Take LCM (Least common factor) on the right hand side of the equation and simplify it.
\[
\sin \theta = \sqrt {\dfrac{{{b^2} - {a^2}}}{{{b^2}}}} \\
\sin \theta = \dfrac{{\sqrt {{b^2} - {a^2}} }}{b}{\text{ }}........{\text{(2)}} \\
\]
(As, square and square-root cancel each other in the denominator)
Now, take the given Left hand side of the equation –
LHS $ = \cos ec\theta + \cot \theta $
Convert the above equation in the terms of $ \sin \theta {\text{ and cos}}\theta $ , where $ \Rightarrow \cos ec\theta = \dfrac{1}{{\sin \theta }}\;{\text{and cot}}\theta {\text{ = }}\dfrac{{\cos \theta }}{{\sin \theta }} $
LHS $ = \dfrac{1}{{\sin \theta }} + \dfrac{{\cos \theta }}{{\sin \theta }} $
Since, the denominator of both the terms are the same, add numerator directly.
LHS $ = \dfrac{{1 + \cos \theta }}{{\sin \theta }} $
Substitute values from the equation $ (1)\;{\text{and (2)}} $
LHS $ = \dfrac{{1 + \dfrac{a}{b}}}{{\dfrac{{\sqrt {{b^2} - {a^2}} }}{b}}} $
Take LCM on the numerator part of the equation on the right
LHS $ = \dfrac{{\dfrac{{b + a}}{b}}}{{\dfrac{{\sqrt {{b^2} - {a^2}} }}{b}}} $
Numerator’s denominator and denominator’s denominator cancel each other.
LHS $ = \dfrac{{b + a}}{{\sqrt {{b^2} - {a^2}} }} $
Using the property of the difference of two squares is - $ (\sqrt {{b^2} - {a^2}} = \sqrt {(b + a)(b - a)} ) $
Also, the square is the product of its square-root into square-root, $ n = \sqrt n \times \sqrt n $
$\Rightarrow$ LHS $ = \dfrac{{\sqrt {(b + a)} \times \sqrt {(b + a)} }}{{\sqrt {(b + a)(b - a)} }} $
$\Rightarrow$ LHS $ = \dfrac{{\sqrt {(b + a)} \times \sqrt {(b + a)} }}{{\sqrt {(b + a)} \times \sqrt {(b - a)} }} $
Same terms from the numerator and the denominator cancel each other.
$\Rightarrow$ LHS $ = \dfrac{{\sqrt {(b + a)} }}{{\sqrt {(b - a)} }} $
$\Rightarrow$ LHS $ = \sqrt {\dfrac{{b + a}}{{b - a}}} $
LHS=RHS
Hence, the given statement is proved.
Note: Remember the basic trigonometric formulas and apply them accordingly. Directly the Pythagoras identity are followed by sines and cosines which states that – $ si{n^2}\theta + co{s^2}\theta = 1 $ and derive other trigonometric functions using it such as tan, cosec, cot and cosec angles.
Complete step-by-step answer:
$ b\cos \theta = a $
Make $ \cos \theta $ the subject –
$ \cos \theta = \dfrac{a}{b}\;{\text{ }}.......{\text{(1)}} $
Using the trigonometric identity that –
$ {\sin ^2}\theta + {\cos ^2}\theta = 1 $
Make the subject
$
\Rightarrow {\sin ^2}\theta = 1 - {\cos ^2}\theta \\
$ \Rightarrow $ \sin \theta = \sqrt {1 - {{\cos }^2}\theta } \\
$
Place the given value in the right hand side of the equation –
$ \Rightarrow \sin \theta = \sqrt {1 - {{\left( {\dfrac{a}{b}} \right)}^2}} $
Simplify the above right hand side of the equation –
\[\Rightarrow \sin \theta = \sqrt {1 - {{\dfrac{a}{{{b^2}}}}^2}} \]
Take LCM (Least common factor) on the right hand side of the equation and simplify it.
\[
\sin \theta = \sqrt {\dfrac{{{b^2} - {a^2}}}{{{b^2}}}} \\
\sin \theta = \dfrac{{\sqrt {{b^2} - {a^2}} }}{b}{\text{ }}........{\text{(2)}} \\
\]
(As, square and square-root cancel each other in the denominator)
Now, take the given Left hand side of the equation –
LHS $ = \cos ec\theta + \cot \theta $
Convert the above equation in the terms of $ \sin \theta {\text{ and cos}}\theta $ , where $ \Rightarrow \cos ec\theta = \dfrac{1}{{\sin \theta }}\;{\text{and cot}}\theta {\text{ = }}\dfrac{{\cos \theta }}{{\sin \theta }} $
LHS $ = \dfrac{1}{{\sin \theta }} + \dfrac{{\cos \theta }}{{\sin \theta }} $
Since, the denominator of both the terms are the same, add numerator directly.
LHS $ = \dfrac{{1 + \cos \theta }}{{\sin \theta }} $
Substitute values from the equation $ (1)\;{\text{and (2)}} $
LHS $ = \dfrac{{1 + \dfrac{a}{b}}}{{\dfrac{{\sqrt {{b^2} - {a^2}} }}{b}}} $
Take LCM on the numerator part of the equation on the right
LHS $ = \dfrac{{\dfrac{{b + a}}{b}}}{{\dfrac{{\sqrt {{b^2} - {a^2}} }}{b}}} $
Numerator’s denominator and denominator’s denominator cancel each other.
LHS $ = \dfrac{{b + a}}{{\sqrt {{b^2} - {a^2}} }} $
Using the property of the difference of two squares is - $ (\sqrt {{b^2} - {a^2}} = \sqrt {(b + a)(b - a)} ) $
Also, the square is the product of its square-root into square-root, $ n = \sqrt n \times \sqrt n $
$\Rightarrow$ LHS $ = \dfrac{{\sqrt {(b + a)} \times \sqrt {(b + a)} }}{{\sqrt {(b + a)(b - a)} }} $
$\Rightarrow$ LHS $ = \dfrac{{\sqrt {(b + a)} \times \sqrt {(b + a)} }}{{\sqrt {(b + a)} \times \sqrt {(b - a)} }} $
Same terms from the numerator and the denominator cancel each other.
$\Rightarrow$ LHS $ = \dfrac{{\sqrt {(b + a)} }}{{\sqrt {(b - a)} }} $
$\Rightarrow$ LHS $ = \sqrt {\dfrac{{b + a}}{{b - a}}} $
LHS=RHS
Hence, the given statement is proved.
Note: Remember the basic trigonometric formulas and apply them accordingly. Directly the Pythagoras identity are followed by sines and cosines which states that – $ si{n^2}\theta + co{s^2}\theta = 1 $ and derive other trigonometric functions using it such as tan, cosec, cot and cosec angles.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)