Answer
Verified
416.7k+ views
Hint: First, the question seems to be wrong. In the given equation, put $x = \dfrac{\pi }{4}$ , we get the left-hand side equal to ${\sec ^2}\dfrac{\pi }{4}\csc \dfrac{\pi }{4} = 2\sqrt 2 $ and the right-hand side equal to ${\sec ^2}\dfrac{\pi }{4} + {\csc ^2}\dfrac{\pi }{4} = 2 + 2 = 4$ . Clearly, the left-hand side is not equal to the right-hand side. The correct equation should be ${\sec ^2}x{\csc ^2}x = {\sec ^2}x + {\csc ^2}x$ . Now to establish this identity we have to show that the part on the right side of the equal to sign is equal to the part on the left side of the equal to sign. For that, we will take any one side and solve it using the trigonometric ratios or identities and make it equal to the other side.
Complete step by step solution:
We have to prove that ${\sec ^2}x{\csc ^2}x = {\sec ^2} + {\csc ^2}x$
We know that $\sec x = \dfrac{1}{{\cos x}}$ and $\csc x = \dfrac{1}{{\sin x}}$
Solving the right-hand side –
$
{\sec ^2}x + {\csc ^2}x = \dfrac{1}{{{{\cos }^2}x}} + \dfrac{1}{{{{\sin }^2}x}} \\
\Rightarrow {\sec ^2}x + {\csc ^2}x = \dfrac{{{{\sin }^2}x + {{\cos }^2}x}}{{{{\sin }^2}x{{\cos }^2}x}} \\
$
We know that ${\sin ^2}x + {\cos ^2}x = 1$ , so we get –
$
\Rightarrow {\sec ^2}x + {\csc ^2}x = \dfrac{1}{{{{\sin }^2}x{{\cos }^2}x}} \\
\Rightarrow {\sec ^2}x + {\csc ^2}x = {\sec ^2}x{\csc ^2}x \\
$
Thus, the right-hand side comes out to be equal to the left-hand side.
Hence proved that ${\sec ^2}x{\csc ^2}x = {\sec ^2}x + {\csc ^2}x$ .
Note: The two sides of a right-angled triangle and one angle other than the right angle are interrelated with each other by trigonometric ratios. The sine of an angle is equal to the ratio of perpendicular and the hypotenuse of the right-angled triangle and the cosine of an angle is equal to the ratio of the base and the hypotenuse of the right-angled triangle. Secant is the reciprocal of the cosine function and cosecant is the reciprocal of the sine function. Identities are used to generalize a relation so that it can be used in other big problems to solve them easily.
Complete step by step solution:
We have to prove that ${\sec ^2}x{\csc ^2}x = {\sec ^2} + {\csc ^2}x$
We know that $\sec x = \dfrac{1}{{\cos x}}$ and $\csc x = \dfrac{1}{{\sin x}}$
Solving the right-hand side –
$
{\sec ^2}x + {\csc ^2}x = \dfrac{1}{{{{\cos }^2}x}} + \dfrac{1}{{{{\sin }^2}x}} \\
\Rightarrow {\sec ^2}x + {\csc ^2}x = \dfrac{{{{\sin }^2}x + {{\cos }^2}x}}{{{{\sin }^2}x{{\cos }^2}x}} \\
$
We know that ${\sin ^2}x + {\cos ^2}x = 1$ , so we get –
$
\Rightarrow {\sec ^2}x + {\csc ^2}x = \dfrac{1}{{{{\sin }^2}x{{\cos }^2}x}} \\
\Rightarrow {\sec ^2}x + {\csc ^2}x = {\sec ^2}x{\csc ^2}x \\
$
Thus, the right-hand side comes out to be equal to the left-hand side.
Hence proved that ${\sec ^2}x{\csc ^2}x = {\sec ^2}x + {\csc ^2}x$ .
Note: The two sides of a right-angled triangle and one angle other than the right angle are interrelated with each other by trigonometric ratios. The sine of an angle is equal to the ratio of perpendicular and the hypotenuse of the right-angled triangle and the cosine of an angle is equal to the ratio of the base and the hypotenuse of the right-angled triangle. Secant is the reciprocal of the cosine function and cosecant is the reciprocal of the sine function. Identities are used to generalize a relation so that it can be used in other big problems to solve them easily.
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 are the Top 10 Largest Countries of the World?
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE