Question

Prove that

(question attached)​

Answer 1

LHS:-

$(1 + \frac{1}{tan {}^{ {}^{2} }x } )(1 + \frac{1}{cot {}^{2}x } )$

$( \frac{tan {}^{2}x + 1 }{tan {}^{2} x} )( \frac{cot {}^{2}x + 1 }{cot {}^{2}x} )$

Now ,

By identity we know

tan²x + 1 = sec²x

cot²x +1 = cosec²x

Putting these values in equation, we get,

$(\frac{sec {}^{2}x }{tan {}^{2}x } )( \frac{cosec {}^{2}x }{cot {}^{2}x } )$

$\frac{sec {}^{2}x \times cosec {}^{2}x }{tan {}^{2}x \times cot {}^{2}x }$

By identity we know,

tan²x × cot²x = 1

Now, putting this value in equation we get,

$sec {}^{2}x \times cosec {}^{2} x$

It can also be written as,

$\frac{1}{cos {}^{2} \: xsin {}^{2}x }$

We know, by identity

sin²x+cos²x = 1

so,

cos²x = 1-sin²x

Putting this value in equation we get,

$\frac{1}{(1 - sin {}^{2}x)sin {}^{2}x }$

$\frac{1}{ - ( sin {}^{4}x - sin {}^{2}x )}$= RHS

Ps:- I have used x in place of theta and also, i can't figure outwhy the question has an extra negative multiplied, which has resulted in alteration of answer. I am pretty much sure my answer has no mistakes.

Anyways, I hope this helped.

Cheers!

Answer 2

Answer:

the trigonometric functions (also called circular functions, angle functions or goniometric functions[1][2]) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis.

Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their side lengths are proportional. Proportionality constants are written within the image: sin θ, cos θ, tan θ, where θ is the common measure of five acute angles.

The most widely used trigonometric functions are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used in modern mathematics. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.[3]

The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extending these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) is often used. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane (from which some isolated points are removed).

Step-by-step explanation:

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