Q) Prove that:

(Q 30 – 30/4/2 – CBSE 2026 Question Paper)

Ans:

[Note: solved in two methods: please choose whichever you find easier]

Method 1:

Step 1: Let’s start from LHS and simplify it:

LHS =

=

=

=

=

=

∵ by trigonometric identity, sin ² θ + cos ² θ = 1

∴ LHS =

∴ LHS =

∴ LHS =

∴ LHS =

∴ LHS =

∴ LHS =

∴ LHS =

∴ LHS = tan x

Step 2: Let’s take RHS & simplify it:

RHS =

=

=

=

=

=

∵ by trigonometric identity, sin ² θ + cos ² θ = 1

∴ RHS =

∴ RHS =

∴ RHS =

∴ RHS =

∴ RHS =

∴ RHS =

∴ RHS =

∴ RHS = tan x

Since LHS = RHS…….. Hence proved!

Method 2:

Given expression is: 

Step 3:

We can rearrange the given terms to make it simpler:

∴

∴  

Now, this is simpler form of given experession and we will try to prove if ts LHS is equal to RHS

Step 4: Let’s start from LHS:

LHS =

=

=

=

=

Step 5: ∵ by trigonometric identity: 1 + tan ² θ = sec ² θ

∴ sec ² θ – tan ² θ = 1

By substituting this value in our LHS expression, we get:

∴ LHS =

= 2 sec x = RHS (of derived expresion in step 3)

Hence Proved !

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