Question

If sin - cos = 0, then what is the value of the expression: (sin4 + cos4)

Answer 1

Correct Question :-

• If sinA - cos A = 0 then Find the value of Sin⁴ A + Cos⁴ A ?

Solution :-

$\qquad$☀We are given equation is Sin A - Cos A = 0

$\pink{\qquad\leadsto\quad \pmb {\mathfrak{Sin A = 0+ Cos A }}}$

$\qquad\leadsto\quad \pmb {\mathfrak{Sin A = Cos A }}$

$\qquad\leadsto\quad \pmb {\mathfrak{ Sin A = 1\times Cos A }}$

$\qquad\leadsto\quad \pmb {\mathfrak{\dfrac{ Sin A }{Cos A }= 1 }}$

$\qquad\leadsto\quad \pmb {\mathfrak{ Tan A =1 }}$

$\qquad\leadsto\quad \pmb {\mathfrak{Tan A = Tan 45° }}$

$\pink{\qquad\leadsto\quad \pmb {\mathfrak{ A = 45° }}}$

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$\blue{\qquad\leadsto\quad \pmb {\mathfrak{Sin⁴ A + Cos⁴ A }}}$

$\qquad\leadsto\quad \pmb {\mathfrak{ (Sin 45°) ⁴ + (Cos 45°)⁴ }}$

$\qquad\leadsto\quad \pmb {\mathfrak{ (\dfrac{1}{√2})⁴ + (\dfrac{1}{√2})⁴ }}$

$\qquad\leadsto\quad \pmb {\mathfrak{ (\dfrac{1}{4}) + (\dfrac{1}{4}) }}$

$\qquad\leadsto\quad \pmb {\mathfrak{\dfrac{ (1+1)}{4} }}$

$\qquad\leadsto\quad \pmb {\mathfrak{\dfrac{ 2}{4} }}$

$\blue{\qquad\leadsto\quad \pmb {\mathfrak{\dfrac{ 1}{2} }}}$

• The value of Sin⁴ A + Cos⁴ A for the given problem is 1/2.

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Answer 2

Answer: ½

Explanation:

Given that,

sinx - cosx = 0

=> sinx = cosx

=> sinx = sin(90° - x)

=> x = 45°.

Now,

sin⁴x + cos⁴x

= (sin45)⁴ + (cos45)⁴

= 2(1/√2)²

= 1/2.

ALITER:

We know,

sin⁴x + cos⁴x = 1 - 2sin²x cos²x

= 1 - 2(1/√2)⁴ = 1 - 2(1/4) = 1/2.

More:

Trigonometry is done using sign convention too. For clockwise it is -ve and for the opposite, it is taken +ve.

Even - odd trig functions:

ODD:

• sin(- x) = - sinx
• cosec(- x) = - cosecx
• tan(-x) = - tanx
• cot(-x) = - cotx.

EVEN:

• cos(- x) = cosx
• sec(- x) = secx.