Answer
397.5k+ views
Hint: Here we are going to use the inverse trigonometric functions relating to secant and cosine. Also, we will use the correlation stating for sine and cosine and then ultimately will derive equations correlating secant and sine angles as per requirement.
Complete step-by-step solution:
Here we will use the reciprocal identity.
Secant and cosine are inverse of each other. Therefore, it can be expressed as –
$\Rightarrow \sec x = \dfrac{1}{{\cos x}}$
Also, by using Pythagorean identity-
$\Rightarrow \cos x = \sin \left( {\dfrac{\pi }{2} - x} \right)$
Also, it can also be expressed as –
$\Rightarrow \cos x = \sqrt {1 - {{\sin }^2}x} $
Hence, the required solution is-
$\Rightarrow \sec x = \dfrac{1}{{\sin \left( {\dfrac{\pi }{2} - x} \right)}}$ or
$\Rightarrow \sec x = \dfrac{1}{{\sqrt {1 - {{\sin }^2}x} }}$
Additional Information: Cosine and secant are inverse functions of each other and sine and cosecant are the inverse functions of each other. The most important property of sines and cosines is that their values lie between minus one and plus one. Every point on the circle is unit circle from the origin. So, the coordinates of any point are within one of zero as well. Directly the Pythagoras identity are followed by sines and cosines which concludes 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.
Also, remember the All STC rule, it is also known as the ASTC rule in geometry. It states that all the trigonometric ratios in the first quadrant ($0^\circ \;{\text{to 90}}^\circ $ ) are positive, sine and cosec are positive in the second quadrant ($90^\circ {\text{ to 180}}^\circ $ ), tan and cot are positive in the third quadrant ($180^\circ \;{\text{to 270}}^\circ $ ) and sin and cosec are positive in the fourth quadrant ($270^\circ {\text{ to 360}}^\circ $ ).
Note: Always remember the correlation between the trigonometric functions and the angles. Also, know the Pythagorean theorem for the angles to get the equivalent values for the different angles and different trigonometric functions.
Complete step-by-step solution:
Here we will use the reciprocal identity.
Secant and cosine are inverse of each other. Therefore, it can be expressed as –
$\Rightarrow \sec x = \dfrac{1}{{\cos x}}$
Also, by using Pythagorean identity-
$\Rightarrow \cos x = \sin \left( {\dfrac{\pi }{2} - x} \right)$
Also, it can also be expressed as –
$\Rightarrow \cos x = \sqrt {1 - {{\sin }^2}x} $
Hence, the required solution is-
$\Rightarrow \sec x = \dfrac{1}{{\sin \left( {\dfrac{\pi }{2} - x} \right)}}$ or
$\Rightarrow \sec x = \dfrac{1}{{\sqrt {1 - {{\sin }^2}x} }}$
Additional Information: Cosine and secant are inverse functions of each other and sine and cosecant are the inverse functions of each other. The most important property of sines and cosines is that their values lie between minus one and plus one. Every point on the circle is unit circle from the origin. So, the coordinates of any point are within one of zero as well. Directly the Pythagoras identity are followed by sines and cosines which concludes 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.
Also, remember the All STC rule, it is also known as the ASTC rule in geometry. It states that all the trigonometric ratios in the first quadrant ($0^\circ \;{\text{to 90}}^\circ $ ) are positive, sine and cosec are positive in the second quadrant ($90^\circ {\text{ to 180}}^\circ $ ), tan and cot are positive in the third quadrant ($180^\circ \;{\text{to 270}}^\circ $ ) and sin and cosec are positive in the fourth quadrant ($270^\circ {\text{ to 360}}^\circ $ ).
Note: Always remember the correlation between the trigonometric functions and the angles. Also, know the Pythagorean theorem for the angles to get the equivalent values for the different angles and different trigonometric functions.
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)