Answer
Verified
418.2k+ views
Hint:The given problem requires us to simplify the given trigonometric expression. The question requires thorough knowledge of trigonometric functions, formulae and identities. The question describes the wide ranging applications of trigonometric identities and formulae. We must keep in mind the trigonometric identities while solving such questions.
Complete step by step answer:
In the given question, we are required to evaluate the value of $\dfrac{{{{\cos }^2}\left( {\dfrac{\pi }{2} - x} \right)}}{{\cos x}}$ using the basic concepts of trigonometry and identities.So, using the trigonometric identity $\cos \left( {\dfrac{\pi }{2} - x} \right) = \sin x$, we get,
$\dfrac{{{{\sin }^2}x}}{{\cos x}}$
Now, we know that $\tan \left( \theta \right)$ is ratio of $\sin \left( \theta \right)$ and $\cos \left( \theta \right)$. So, replacing $\dfrac{{\sin \theta }}{{\cos \theta }}$ by $\tan \left( \theta \right)$, we get,
$\tan \left( x \right)\sin \left( x \right)$
Hence, we get the value of trigonometric expression $\dfrac{{{{\cos }^2}\left( {\dfrac{\pi }{2} - x} \right)}}{{\cos x}}$ as $\tan \left( x \right)\sin \left( x \right)$.
Additional Information:
There are six trigonometric ratios: $\sin \theta $, $\cos \theta $, $\tan \theta $, $\cos ec\theta $, $\sec \theta $and $\cot \theta $. $\cos ec\theta $ is reciprocal of $\sin \theta $. Similarly, $\sec \theta $ is reciprocal of $\cos \theta $. $\tan \theta $ is ratio of $\sin \theta $ to $\cos \theta $ . Also,$\cot \theta $ is the reciprocal of $\tan \theta $. Hence, $\cot \theta $ is the ratio of $\cos \theta $ to $\sin \theta $ . Basic trigonometric identities include ${\sin ^2}\theta + {\cos ^2}\theta = 1$, ${\sec ^2}\theta = {\tan ^2}\theta + 1$ and $\cos e{c^2}\theta = {\cot ^2}\theta + 1$. These identities are of vital importance for solving any question involving trigonometric functions and identities. All the trigonometric ratios can be converted into each other using the simple trigonometric identities listed above.
Note: The given problem involves the use of trigonometric formulae and identities. Such questions require thorough knowledge of trigonometric conversions and ratios. Algebraic operations and rules like transposition rule come into significant use while solving such problems.
Complete step by step answer:
In the given question, we are required to evaluate the value of $\dfrac{{{{\cos }^2}\left( {\dfrac{\pi }{2} - x} \right)}}{{\cos x}}$ using the basic concepts of trigonometry and identities.So, using the trigonometric identity $\cos \left( {\dfrac{\pi }{2} - x} \right) = \sin x$, we get,
$\dfrac{{{{\sin }^2}x}}{{\cos x}}$
Now, we know that $\tan \left( \theta \right)$ is ratio of $\sin \left( \theta \right)$ and $\cos \left( \theta \right)$. So, replacing $\dfrac{{\sin \theta }}{{\cos \theta }}$ by $\tan \left( \theta \right)$, we get,
$\tan \left( x \right)\sin \left( x \right)$
Hence, we get the value of trigonometric expression $\dfrac{{{{\cos }^2}\left( {\dfrac{\pi }{2} - x} \right)}}{{\cos x}}$ as $\tan \left( x \right)\sin \left( x \right)$.
Additional Information:
There are six trigonometric ratios: $\sin \theta $, $\cos \theta $, $\tan \theta $, $\cos ec\theta $, $\sec \theta $and $\cot \theta $. $\cos ec\theta $ is reciprocal of $\sin \theta $. Similarly, $\sec \theta $ is reciprocal of $\cos \theta $. $\tan \theta $ is ratio of $\sin \theta $ to $\cos \theta $ . Also,$\cot \theta $ is the reciprocal of $\tan \theta $. Hence, $\cot \theta $ is the ratio of $\cos \theta $ to $\sin \theta $ . Basic trigonometric identities include ${\sin ^2}\theta + {\cos ^2}\theta = 1$, ${\sec ^2}\theta = {\tan ^2}\theta + 1$ and $\cos e{c^2}\theta = {\cot ^2}\theta + 1$. These identities are of vital importance for solving any question involving trigonometric functions and identities. All the trigonometric ratios can be converted into each other using the simple trigonometric identities listed above.
Note: The given problem involves the use of trigonometric formulae and identities. Such questions require thorough knowledge of trigonometric conversions and ratios. Algebraic operations and rules like transposition rule come into significant use while solving such problems.
Recently Updated Pages
what is the correct chronological order of the following class 10 social science CBSE
Which of the following was not the actual cause for class 10 social science CBSE
Which of the following statements is not correct A class 10 social science CBSE
Which of the following leaders was not present in the class 10 social science CBSE
Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE
Which one of the following places is not covered by class 10 social science CBSE
Trending doubts
Which places in India experience sunrise first and class 9 social science CBSE
Which are the Top 10 Largest Countries of the World?
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Give 10 examples for herbs , shrubs , climbers , creepers
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE