Question

$\sf{The \:value\:of\:sec² \: (tan^{-1})\:+\: cosec² \: (cot^{-1} \: 3) \: is}$
Options are::-

a) 5
b) 10
c) 15
d) 20​

Answer 1

Answer:

My answer is 12

since number after tan inverse was missing i took 1

if its other number just replace with one and u will get the answer

Answer 2

★ Concept :-

Here the concept of Inverse Trignometry has been used. We see that we are given an equation where we need to find the value of the equation. The simplest way to solve such equations is by reducing the terms. Here firstly we shall reduce each term and then find the final answer.

Let's do it !!

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★ Solution :-

Given,

» sec² (tan¯¹ 1) + cosec² (cot¯¹ 3)

Here we need to find the value of this equation.

We know that,

• sec² x = 1 + tan² x

• Here x = tan¯¹ 1

So by applying this, we get

>> [1 + tan² (tan¯¹ 1) ] + cosec² (cot¯¹ 3)

>> 1 + [tan (tan¯¹ 1)]² + cosec² (cot¯¹ 3)

Here tan and tan¯¹ will cancel each other giving 1.

>> 1 + [1]² + cosec² (cot¯¹ 3)

>> 1 + 1 + cosec² (cot¯¹ 3)

>> 2 + cosec² (cot¯¹ 3)

We know that,

• cosec² y = 1 + cot² y

• Here x = cot¯¹ 3

By applying this here, we get

>> 2 + [ 1 + cot² (cot¯¹ 3) ]

>> 2 + 1 + [cot (cot¯¹ 3)]²

Here cot and cot¯¹ will cancel each other giving 1. So,

>> 2 + 1 + [(3)]²

>> 3 + 9

>> 12

This is the required answer.

→ Hence, required value = 12

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★ More to know :-

• sin² x + cos² x = 1

• sin x = cos(90° - x)

• cos x = sin(90° - x)

• tan x = cot(90° - x)

• cot x = tan(90° - x)

• cosec x = sec(90° - x)

• sec x = cosec(90° - x)