if tan theta = 5/12, show that tan^2 theta - sin^2 theta = Sin^4 theta sec^2 theta
Answers
Step-by-step explanation:
Given :-
Tan θ = 5/12
To find :-
Show that Tan² θ - Sin² θ = Sin⁴ θ Sec² θ
Solution :-
Given that
Tan θ = 5/12
=> 1/Tan θ = 1/(5/12)
=> Cot θ = 12/5
On squaring both sides then
=> (Cot θ)² = (12/5)²
=> Cot² θ = 144/25
On adding 1 both sides then
=> 1+ Cot² θ = 1+(144/25)
=> 1+Cot² θ = (25+144)/25
=> Cosec² θ = 169/25
Since Cosec² θ - Cot² θ = 1
=> 1/Sin² θ = 169/25
=> Sin² θ = 25/169
=> 1-Sin² θ = 1-(25/169)
=> 1-Sin² θ = (169-25)/169
=> 1- Sin² θ = 144/169
=> Cos² θ = 144/169
Since Sin² A + Cos² A = 1
=> Cos θ = √(144/169)
=> Cos θ = 12/13
=> 1/Cos θ = 1/(12/13)
=> Sec θ = 13/12
=> Sec² θ = (13/12)²
=> Sec² θ = 169/144
Now
On taking LHS
Tan² θ - Sin² θ
=> (25/144) - (25/169)
=> 25[(1/144)-(1/169)]
=> 25[(169-144)/24336]
=> 25(25/24336)
=> (25×25)/(24336)
=> 625/24336 ------------------(1)
On taking RHS
Sin⁴ θ Sec² θ
=> (25/169)² ×(169/144)
=> (625×169)/(169×169×144)
=>625/(169×144)
=> 625/24336 -----------------(2)
From (1) &(2)
LHS = RHS
Tan² θ - Sin² θ = Sin⁴ θ Sec² θ
Hence, Proved.
Used formulae:-
→ Cosec² A - Cot² A = 1
→ Sin² A + Cos² A = 1
→ Cosec A = 1/Sin A
→ Sec A = 1/Cos A
→ Cot A = 1/Tan A
Step-by-step explanation:
tan theta = 5/12
sec2 theta = 1 + tan2 theta
= 1 + 25/144 = 144+25/144 =
44
4.0
Math
10 points
Vol 86
169/144
sec theta = 13/12
cos theta = 1/ sec theta = 12/13
cos2 theta + sin2 theta =1
sin2 theta = 1- cos2 theta
= 1- 144/169 = 169-144/169 =
25/169
sin theta = root of 25/ 169 = 5/13