How to prove 5secx+4 =9secsquarex- 4tansquarex?

#5sec^2x+4 =9sec^2x- 4tan^2x?#

3 Answers
Jun 5, 2018

Please find a Proof in Explanation.

Explanation:

I hope, the Question is to prove :

#5sec^2x+4 =9sec^2x- 4tan^2x.#

If we use the Identity : # tan^2x=sec^2x-1#, the Problem

can be easily proved as follows :

#9sec^2x-4tan^2x#,

#=9sec^2x-4(sec^2x-1)#,

#=9sec^2x-4sec^2x+4#,

#=5sec^2x+4#, as desired!

Otherwise, #9sec^2x-4tan^2x#,

#=9sec^2x-4{(sin^2x)/cos^2x}#,

#=9sec^2x-4{(1-cos^2x)/cos^2x}#,

#=9sec^2x-4{1/cos^2x-cos^2x/cos^2x}#,

#=9sec^2x-4{sec^2x-1}#, and the rest is as above.

Jun 5, 2018

The question is in error, and in fact

# 5sec^2x+4 -= 9sec^2x-4tan^2x #

Explanation:

The question is in error, and in fact

# 5sec^2x+4 -= 9sec^2x-4tan^2x #

As can be shown, if we consider the LHS:

# LHS -= 5sec^2x+4 #

# \ \ \ \ \ \ \ \ = 5(1+tan^2x)+4 #

# \ \ \ \ \ \ \ \ = 5+5tan^2x+4 #

# \ \ \ \ \ \ \ \ = 9+5tan^2x #

# \ \ \ \ \ \ \ \ = 9+5tan^2x + 4tan^2x - 4tan^2x#

# \ \ \ \ \ \ \ \ = 9+9tan^2x - 4tan^2x#

# \ \ \ \ \ \ \ \ = 9(1+tan^2x) - 4tan^2x#

# \ \ \ \ \ \ \ \ = 9sec^2x - 4tan^2x#

# \ \ \ \ \ \ \ \ -= RHS \ \ \ \ \ # QED

Jun 5, 2018

Answer for the question :
#5color(red)(sec^2x)+4=9sec^2x-4tan^2x#
Please see below

Explanation:

We know that,

#color(blue)(sec^2theta-tan^2theta=1=>sec^2theta- 1=tan^2theta#

We have to prove,

#5sec^2x+4=9sec^2x-4tan^2x#

We take,

#RHS=9sec^2x-4tan^2x#

#color(white)(RHS)=9sec^2x-4(sec^2x-1)#

#color(white)(RHS)=9sec^2x-4sec^2x+4#

#color(white)(RHS)=5sec^2x+4#

#RHS=LHS#