Explanation

Identity: 1 + tan²θ = sec²θ

Last verified on 04-05-2026

Explanation

sin²θ + cos²θ = 1  

(3/5)² + cos²θ = 1  

9/25 + cos²θ = 1  

cos²θ = 1 - 9/25  

cos²θ = 16/25  

cosθ = ±4/5  

If θ is acute, cosθ = 4/5

Last verified on 04-05-2026

Explanation

Right triangle: a² + b² = c²  

5² + b² = 13²  

25 + b² = 169  

b² = 169 - 25  

b² = 144  

b = 12  

Last verified on 04-05-2026

Explanation

Third side rule: difference < x < sum  

10 - 7 < x < 10 + 7  

3 < x < 17

16 → 3 < 16 < 17, possible

Last verified on 04-05-2026

Explanation

5/[x(x + 2)] = A/x + B/(x + 2)  

Multiply both sides by x(x + 2)  

5 = A(x + 2) + Bx  

Put x = 0  

5 = A(0 + 2) + B×0  

5 = 2A  

A = 5/2  

Put x = -2  

5 = A(-2 + 2) + B(-2)  

5 = -2B  

B = -5/2  

So 5/[x(x + 2)] = 5/(2x) - 5/[2(x + 2)]

Last verified on 04-05-2026

Explanation

Let y = 2x + 1  

Swap x and y  

x = 2y + 1  

Solve for y  

x - 1 = 2y  

y = (x - 1) / 2  

f⁻¹(x) = (x - 1) / 2

Last verified on 04-05-2026

Explanation

y = k × x²  

Put y = 36, x = 3  

36 = k × 3²  

36 = k × 9  

k = 4

Now y = 4 × x²  

Put x = 5  

y = 4 × 5²  

y = 4 × 25

y = 100

Last verified on 04-05-2026

Explanation

Solve: (2x + 3) / (x - 1) = 3

2x + 3 = 3(x - 1)  

2x + 3 = 3x - 3  

3 + 3 = 3x - 2x  

x = 6

Last verified on 04-05-2026

Explanation

Old equation: x² - 5x + 6 = 0  

Old roots: (x - 2)(x - 3) = 0 → roots are 2 and 3  

Add 2 to each root  

New roots: 2 + 2 = 4 and 3 + 2 = 5  

New equation: (x - 4)(x - 5) = 0  

x² - 5x - 4x + 20 = 0  

x² - 9x + 20 = 0

Last verified on 04-05-2026

Explanation

Two consecutive integers that add to 7 are 3 and 4  

3 + 4 = 7  

3 × 4 = 12  

So factors are (x - 3)(x - 4) = x² - 7x + 12  

k =  12

Last verified on 04-05-2026