if (sin x + cosec x)^2 + (sec x + cos x)2= k + tan^2 x + cot^2 x then k=
Answers
Step-by-step explanation:
Given :-
(Sin x+Cosec x)²+(Sec x+Cos x)² = k+Tan² x+Cot² x
To find :-
Find the value of k ?
Solution :-
Given equation is
(Sin x+Cosec x)²+(Sec x+Cos x)²
= k+Tan² x+Cot² x
On taking LHS :
(Sin x+Cosec x)²+(Sec x+Cos x)²
=>( Sin² x+ Cosec² x + 2 Sin x Cosec x ) + ( Sec² x + Cos² x + 2Sec x Cos x)
Since (a+b)² = a²+2ab+b²
=> (Sin² x+Cos²x) +Cosec²x+Sec²x
+[2Sin x(1/Sin x) ]+ [2 Cos x (1/Cos x)]
=> (Sin² x+Cos²x) +Cosec²x+Sec²x +
[2 (Sin x/Sin x) ]+ [2 (Cos x /Cos x)]
=> 1 + Cosec² x+ Sec² x +2(1)+2(1)
Since Sin² A+ Cos² A = 1
=> 1+ Cosec² x+ Sec² x +2+2
=>(1+2+2)+Cosec² x+ Sec² x
=> 5+Cosec² x+ Sec² x
We know that
Cosec²x-Cot² x = 1 and
Sec² x - Tan² x = 1
=> 5+ 1+Cot² x + 1+ Tan² x
=> (5+1+1)+Tan² x+ Cot² x
=> 7 + Tan² x + Cot² x
Now given equation becomes
=> 7 + Tan² x + Cot² x = k+Tan² x+Cot² x
=> 7 + Tan² x + Cot² x -Tan² x-Cot² x = k
=> 7 = k
=> k = 7
Therefore, k = 7
Answer:-
The value of k for the given problem is 7
Used formulae:-
- (a+b)² = a²+2ab+b²
- Cosec²x-Cot² x = 1
- Sec² x - Tan² x = 1
- Sec x = 1/Cos x
- Cosec x = 1/Sin x